Massive Gravity - Springer · of massive gravity emerge from a higher-dimensional theory of general...

Living Rev. Relativity, 17, (2014), 7http://www.livingreviews.org/lrr-2014-7

doi:10.12942/lrr-2014-7

Massive Gravity

Claudia de RhamCERCA & Physics DepartmentCase Western Reserve University

10900 Euclid Ave, Cleveland, OH 44106, USAemail: [email protected]

http://www.phys.cwru.edu/~claudia/

Accepted: 18 July 2014Published: 25 August 2014

Abstract

We review recent progress in massive gravity. We start by showing how different theoriesof massive gravity emerge from a higher-dimensional theory of general relativity, leading tothe Dvali–Gabadadze–Porrati model (DGP), cascading gravity, and ghost-free massive gravity.We then explore their theoretical and phenomenological consistency, proving the absence ofBoulware–Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solu-tions in these models. Finally, we present alternative and related models of massive gravitysuch as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Keywords: General relativity, Gravity, Alternative theories of gravity, Modified theories ofgravity, Massive gravity

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Contents

1 Introduction 7

2 Massive and Interacting Fields 102.1 Proca field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Maxwell kinetic term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Proca mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Abelian Higgs mechanism for electromagnetism . . . . . . . . . . . . . . . . 132.1.4 Interacting spin-1 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Spin-2 field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Einstein–Hilbert kinetic term . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Fierz–Pauli mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Van Dam–Veltman–Zakharov discontinuity . . . . . . . . . . . . . . . . . . 18

2.3 From linearized diffeomorphism to full diffeomorphism invariance . . . . . . . . . . 202.4 Non-linear Stuckelberg decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Boulware–Deser ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

I Massive Gravity from Extra Dimensions 26

3 Higher-Dimensional Scenarios 26

4 The Dvali–Gabadadze–Porrati Model 274.1 Gravity induced on a brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 Perturbations about flat spacetime . . . . . . . . . . . . . . . . . . . . . . . 284.1.2 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Brane-bending mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Phenomenology of DGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Friedmann equation in de Sitter . . . . . . . . . . . . . . . . . . . . . . . . 334.3.2 General Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3.3 Observational viability of DGP . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Self-acceleration branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Degravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5.1 Cascading gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Deconstruction 425.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Metric versus Einstein–Cartan formulation of GR . . . . . . . . . . . . . . . 425.1.2 Gauge-fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1.3 Discretization in the vielbein . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Ghost-free massive gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.1 Simplest discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.2 Generalized mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Multi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 Bi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Coupling to matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 No new kinetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

II Ghost-free Massive Gravity 53

6 Massive, Bi- and Multi-Gravity Formulation: A Summary 53

7 Evading the BD Ghost in Massive Gravity 567.1 ADM formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.1.1 ADM formalism for GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1.2 ADM counting in massive gravity . . . . . . . . . . . . . . . . . . . . . . . . 587.1.3 Eliminating the BD ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2 Absence of ghost in the Stuckelberg language . . . . . . . . . . . . . . . . . . . . . 637.2.1 Physical degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2.2 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.2.3 Full proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.2.4 Stuckelberg method on arbitrary backgrounds . . . . . . . . . . . . . . . . . 66

7.3 Absence of ghost in the vielbein formulation . . . . . . . . . . . . . . . . . . . . . . 687.4 Absence of ghosts in multi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Decoupling Limits 718.1 Scaling versus decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.2 Massive gravity as a decoupling limit of bi-gravity . . . . . . . . . . . . . . . . . . 73

8.2.1 Minkowski reference metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.2.2 (A)dS reference metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.2.3 Arbitrary reference metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.3 Decoupling limit of massive gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 768.3.1 Interaction scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768.3.2 Operators below the scale Λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . 768.3.3 Λ3-decoupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3.4 Vector interactions in the Λ3-decoupling limit . . . . . . . . . . . . . . . . . 808.3.5 Beyond the decoupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3.6 Decoupling limit on (Anti) de Sitter . . . . . . . . . . . . . . . . . . . . . . 82

8.4 Λ3-decoupling limit of bi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9 Extensions of Ghost-free Massive Gravity 899.1 Mass-varying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.2 Quasi-dilaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.2.2 Extended quasi-dilaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.3 Partially massless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.3.1 Motivations behind PM gravity . . . . . . . . . . . . . . . . . . . . . . . . . 949.3.2 The search for a PM theory of gravity . . . . . . . . . . . . . . . . . . . . . 95

10 Massive Gravity Field Theory 9710.1 Vainshtein mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

10.1.1 Effective coupling to matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.1.2 Static and spherically symmetric configurations in Galileons . . . . . . . . . 9910.1.3 Static and spherically symmetric configurations in massive gravity . . . . . 101

10.2 Validity of the EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10210.3 Non-renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.4 Quantum corrections beyond the decoupling limit . . . . . . . . . . . . . . . . . . . 105

10.4.1 Matter loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.4.2 Graviton loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.5 Strong coupling scale vs cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10710.6 Superluminalities and (a)causality . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

10.6.1 Superluminalities in Galileons . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.6.2 Superluminalities in massive gravity . . . . . . . . . . . . . . . . . . . . . . 111

10.6.3 Superluminalities vs Boulware–Deser ghost vs Vainshtein . . . . . . . . . . 115

10.7 Galileon duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

III Phenomenological Aspects of Ghost-free Massive Gravity 118

11 Phenomenology 118

11.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

11.1.1 Speed of propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

11.1.2 Additional polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

11.2 Solar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

11.3 Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

11.4 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

11.5 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

12 Cosmology 128

12.1 Cosmology in the decoupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

12.2 FLRW solutions in the full theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

12.2.1 Absence of flat/closed FLRW solutions . . . . . . . . . . . . . . . . . . . . . 131

12.2.2 Open FLRW solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

12.3 Inhomogenous/anisotropic cosmological solutions . . . . . . . . . . . . . . . . . . . 133

12.3.1 Special isotropic and inhomogeneous solutions . . . . . . . . . . . . . . . . . 133

12.3.2 General anisotropic and inhomogeneous solutions . . . . . . . . . . . . . . . 136

12.4 Massive gravity on FLRW and bi-gravity . . . . . . . . . . . . . . . . . . . . . . . . 136

12.4.1 FLRW reference metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

12.4.2 Bi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

12.5 Other proposals for cosmological solutions . . . . . . . . . . . . . . . . . . . . . . . 139

IV Other Theories of Massive Gravity 141

13 New Massive Gravity 141

13.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

13.2 Absence of Boulware–Deser ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

13.3 Decoupling limit of new massive gravity . . . . . . . . . . . . . . . . . . . . . . . . 143

13.4 Connection with bi-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

13.5 3D massive gravity extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

13.6 Other 3D theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

13.6.1 Topological massive gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

13.6.2 Supergravity extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

13.6.3 Critical gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

13.7 Black holes and other exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 148

13.8 New massive gravity holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

13.9 Zwei-dreibein gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

14 Lorentz-Violating Massive Gravity 15114.1 SO(3)-invariant mass terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15114.2 Phase 𝑚1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

14.2.1 Degrees of freedom on Minkowski . . . . . . . . . . . . . . . . . . . . . . . . 15314.2.2 Non-perturbative degrees of freedom . . . . . . . . . . . . . . . . . . . . . . 153

14.3 General massive gravity (𝑚0 = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15414.3.1 First explicit Lorentz-breaking example with five dofs . . . . . . . . . . . . 15414.3.2 Second example of Lorentz-breaking with five dofs . . . . . . . . . . . . . . 15514.3.3 Absence of vDVZ and strong coupling scale . . . . . . . . . . . . . . . . . . 15514.3.4 Cosmology of general massive gravity . . . . . . . . . . . . . . . . . . . . . 156

15 Non-local massive gravity 157

16 Outlook 159

References 160

Massive Gravity 7

1 Introduction

For almost a century, the theory of general relativity (GR) has been known to describe the force ofgravity with impeccable agreement with observations. Despite all the successes of GR the searchfor alternatives has been an ongoing challenge since its formulation. Far from a purely academicexercise, the existence of consistent alternatives to describe the theory of gravitation is actuallyessential to test the theory of GR. Furthermore, the open questions that remain behind the puzzlesat the interface between gravity/cosmology and particle physics such as the hierarchy problem, theold cosmological constant problem and the origin of the late-time acceleration of the Universe havepushed the search for alternatives to GR.

While it was not formulated in this language at the time, from a more modern particle physicsperspective GR can be thought of as the unique theory of a massless spin-2 particle [287, 483, 175,225, 76], and so in order to find alternatives to GR one should break one of the underlying assump-tions behind this uniqueness theorem. Breaking Lorentz invariance and the notion of spin alongwith it is probably the most straightforward since non-Lorentz invariant theories include a greatamount of additional freedom. This possibility has been explored at length in the literature; seefor instance [398] for a review. Nevertheless, Lorentz invariance is observationally well constrainedby both particle and astrophysics. Another possibility is to maintain Lorentz invariance and thenotion of spin that goes with it but to consider gravity as being the representation of a higherspin. This idea has also been explored; see for instance [466, 52] for further details. In this review,we shall explore yet another alternative: Maintaining the notion that gravity is propagated by aspin-2 particle but considering this particle to be massive. From the particle physics perspective,this extension seems most natural since we know that the particles carrier of the electroweak forceshave to acquire a mass through the Higgs mechanism.

Giving a mass to a spin-2 (and spin-1) field is an old idea and in this review we shall summarizethe approach of Fierz and Pauli, which dates back to 1939 [226]. While the theory of a massive spin-2 field is in principle simple to derive, complications arise when we include interactions betweenthis spin-2 particle and other particles as should be the case if the spin-2 field is to describe thegraviton.

At the linear level, the theory of a massless spin-2 field enjoys a linearized diffeomorphism (diff)symmetry, just as a photon enjoys a 𝑈(1) gauge symmetry. But unlike for a photon, coupling thespin-2 field with external matter forces this symmetry to be realized in a different way non-linearly.As a result, GR is a fully non-linear theory, which enjoys non-linear diffeomorphism invariance(also known as general covariance or coordinate invariance). Even though this symmetry is brokenwhen dealing with a massive spin-2 field, the non-linearities are inherited by the field. So, unlike asingle isolated massive spin-2 field, a theory of massive gravity is always fully non-linear (and as aconsequence non-renormalizable) just as for GR. The fully non-linear equivalent to GR for massivegravity has been a much more challenging theory to obtain. In this review we will summarize afew different approaches to deriving consistent theories of massive gravity and will focus on recentprogress. See Ref. [309] for an earlier review on massive gravity, as well as Refs. [134] and [336] forother reviews relating Galileons and massive gravity.

When dealing with a theory of massive gravity two elements have been known to be problematicsince the seventies. First, a massive spin-2 field propagates five degrees of freedom no matter howsmall its mass. At first this seems to suggest that even in the massless limit, a theory of massivegravity could never resemble GR, i.e., a theory of a massless spin-2 field with only two propagatingdegrees of freedom. This subtlety is at the origin of the vDVZ discontinuity (van Dam–Veltman–Zakharov [465, 497]). The resolution behind that puzzle was provided by Vainshtein two yearslater and lies in the fact that the extra degree of freedom responsible for the vDVZ discontinuitygets screened by its own interactions, which dominate over the linear terms in the massless limit.This process is now relatively well understood [463] (see also Ref. [35] for a recent review). The

Living Reviews in Relativityhttp://www.livingreviews.org/lrr-2014-7


8 Claudia de Rham

Vainshtein mechanism also comes hand in hand with its own set of peculiarities like strong couplingand superluminalities, which we will discuss in this review.

A second element of concern in dealing with a theory of massive gravity is the realization thatmost non-linear extensions of Fierz–Pauli massive gravity are plagued with a ghost, now knownas the Boulware–Deser (BD) ghost [75]. The past decade has seen a revival of interest in massivegravity with the realization that this BD ghost could be avoided either in a model of soft massivegravity (not a single massive pole for the graviton but rather a resonance) as in the DGP (Dvali–Gabadadze–Porrati) model or its extensions [208, 209, 207], or in a three-dimensional model ofmassive gravity as in ‘new massive gravity’ (NMG) [66] or more recently in a specific ghost-freerealization of massive gravity (also known as dRGT in the literature) [144].

With these developments several new possibilities have become a reality:

∙ First, one can now more rigorously test massive gravity as an alternative to GR. We willsummarize the different phenomenologies of these models and their theoretical as well asobservational bounds through this review. Except in specific cases, the graviton mass is typ-ically bounded to be a few times the Hubble parameter today, that is 𝑚 . 10−30 – 10−33 eVdepending on the exact models. In all of these models, if the graviton had a mass muchsmaller than 10−33 eV, its effect would be unseen in the observable Universe and such amass would thus be irrelevant. Fortunately there is still to date an open window of oppor-tunity for the graviton mass to be within an interesting range and providing potentially newobservational signatures.

∙ Second, these developments have opened up the door for theories of interacting metrics, asuccess long awaited. Massive gravity was first shown to be expressible on an arbitrary refer-ence metric in [296]. It was then shown that the reference metric could have its own dynamicsleading to the first consistent formulation of bi-gravity [293]. In bi-gravity two metrics areinteracting and the mass spectrum is that of a massless spin-2 field interacting with a massivespin-2 field. It can, therefore, be seen as the theory of general relativity interacting (fullynon-linearly) with a massive spin-2 field. This is a remarkable new development in both fieldtheory and gravity.

∙ The formulation of massive gravity and bi-gravity in the vielbein language were shown tobe both analytic and much more natural and allowed for a general formulation of multi-gravity [314] where an arbitrary number of spin-2 fields may interact together.

∙ Finally, still within the theoretical progress front, all of these successes provided full anddefinite proof for the absence of Boulware–Deser ghosts in these types of theories; see [295],which has then been translated into a multitude of other languages. This also opens the doorfor new types of theories that can propagate fewer degrees than naively thought.

Independent of this, developments in massive gravity, bi-gravity and multi-gravity have also openedup new theoretical avenues, which we will summarize, and these remain very much an active areaof progress. On the phenomenological front, a genuine task force has been devoted to finding bothexact and approximate solutions in these types of gravitational theories, including the ones relevantfor black holes and for cosmology. We shall summarize these in the review.

This review is organized as follows: We start by setting the formalism for massive and masslessspin-1 and -2 fields in Section 2 and emphasize the Stuckelberg language both for the Proca andthe Fierz–Pauli fields. In Part I we then derive consistent theories using a higher-dimensionalframework, either using a braneworld scenario a la DGP in Section 4, or via a discretization1 (orKaluza–Klein reduction) of the extra dimension in Section 5. This second approach leads to the

1 In the case of ‘discretization’ or ‘deconstruction’ the higher dimensional approach has been useful only ‘afterthe fact’. Unlike for DGP, in massive gravity the extra dimension is purely used as a mathematical tool and the



Massive Gravity 9

theory of ghost-free massive gravity (also known as dRGT) which we review in more depth inPart II. Its formulation is summarized in Section 6, before tackling other interesting aspects suchas the fate of the BD ghost in Section 7, deriving its decoupling limit in Section 8, and variousextensions in Section sec:Extensions. The Vainshtein mechanism and other related aspects arediscussed in Section 10. The phenomenology of ghost-free massive gravity is then reviewed inPart III including a discussion of solar-system tests, gravitational waves, weak lensing, pulsars,black holes and cosmology. We then conclude with other related theories of massive gravity inPart IV, including new massive gravity, Lorentz breaking theories of massive gravity and non-localversions.

Notations and conventions: Throughout this review, we work in units where the reducedPlanck constant ~ and the speed of light 𝑐 are set to unity. The gravitational Newton constant𝐺𝑁 is related to the Planck scale by 8𝜋𝐺𝑁 = 𝑀−2

Pl . Unless specified otherwise, 𝑑 represents thenumber of spacetime dimensions. We use the mainly + convention (− + · · ·+) and space indicesare denoted by 𝑖, 𝑗, · · · = 1, · · · , 𝑑− 1 while 0 represents the time-like direction, 𝑥0 = 𝑡.

We also use the symmetric convention: (𝑎, 𝑏) = 12 (𝑎𝑏+ 𝑏𝑎) and [𝑎, 𝑏] = 1

2 (𝑎𝑏− 𝑏𝑎). Throughoutthis review, square brackets of a tensor indicates the trace of tensor, for instance [X] = X𝜇𝜇, [X2] =X𝜇𝜈X𝜈𝜇, etc. . . . We also use the notation Π𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋, and I = 𝛿𝜇𝜈 . 𝜀𝜇𝜈𝛼𝛽 and 𝜀𝐴𝐵𝐶𝐷𝐸 representthe Levi-Cevita symbol in respectively four and five dimensions, 𝜀0123 = 𝜀01234 = 1 = 𝜀0123.

formulation of the theory was first performed in four dimensions. In this case massive gravity is not derived perse from the higher-dimensional picture but rather one can see how the structure of general relativity in higherdimensions is tied to that of the mass term.



10 Claudia de Rham

2 Massive and Interacting Fields

2.1 Proca field

2.1.1 Maxwell kinetic term

Before jumping into the subtleties of massive spin-2 field and gravity in general, we start thisreview with massless and massive spin-1 fields as a warm up. Consider a Lorentz vector field 𝐴𝜇living on a four-dimensional Minkowski manifold. We focus this discussion to four dimensions andthe extension to 𝑑 dimensions is straightforward. Restricting ourselves to Lorentz invariant andlocal actions for now, the kinetic term can be decomposed into three possible contributions:

ℒspin−1kin = 𝑎1ℒ1 + 𝑎2ℒ2 + 𝑎3ℒ3 , (2.1)

where 𝑎1,2,3 are so far arbitrary dimensionless coefficients and the possible kinetic terms are givenby

ℒ1 = 𝜕𝜇𝐴𝜈𝜕𝜇𝐴𝜈 (2.2)

ℒ2 = 𝜕𝜇𝐴𝜇𝜕𝜈𝐴

𝜈 (2.3)

ℒ3 = 𝜕𝜇𝐴𝜈𝜕𝜈𝐴

𝜇 , (2.4)

where in this section, indices are raised and lowered with respect to the flat Minkowski metric.The first and third contributions are equivalent up to a boundary term, so we set 𝑎3 = 0 withoutloss of generality.

We now proceed to establish the behavior of the different degrees of freedom (dofs) present inthis theory. A priori, a Lorentz vector field 𝐴𝜇 in four dimensions could have up to four dofs,which we can split as a transverse contribution 𝐴⊥

𝜇 satisfying 𝜕𝜇𝐴⊥𝜇 = 0 bearing a priori three dofs

and a longitudinal mode 𝜒 with 𝐴𝜇 = 𝐴⊥𝜇 + 𝜕𝜇𝜒.

Helicity-0 mode

Focusing on the longitudinal (or helicity-0) mode 𝜒, the kinetic term takes the form

ℒ𝜒kin = (𝑎1 + 𝑎2)𝜕𝜇𝜕𝜈𝜒𝜕𝜇𝜕𝜈𝜒 = (𝑎1 + 𝑎2)(2𝜒)

2 , (2.5)

where 2 = 𝜂𝜇𝜈𝜕𝜇𝜕𝜈 represents the d’Alembertian in flat Minkowski space and the second equalityholds after integrations by parts. We directly see that unless 𝑎1 = −𝑎2, the kinetic term for thefield 𝜒 bears higher time (and space) derivatives. As a well known consequence of Ostrogradsky’stheorem [421], two dofs are actually hidden in 𝜒 with an opposite sign kinetic term. This can beseen by expressing the propagator 2−2 as the sum of two propagators with opposite signs:

1

22= lim𝑚→0

1

2𝑚2

(1

2−𝑚2− 1

2+𝑚2

), (2.6)

signaling that one of the modes always couples the wrong way to external sources. The mass 𝑚 ofthis mode is arbitrarily low which implies that the theory (2.1) with 𝑎3 = 0 and 𝑎1+𝑎2 = 0 is alwayssick. Alternatively, one can see the appearance of the Ostrogradsky instability by introducing aLagrange multiplier ��(𝑥), so that the kinetic action (2.5) for 𝜒 is equivalent to

ℒ𝜒kin = (𝑎1 + 𝑎2)

(��2𝜒− 1

4��2

), (2.7)



Massive Gravity 11

after integrating out the Lagrange multiplier2 �� ≡ 22𝜒. We can now perform the change ofvariables 𝜒 = 𝜑1 + 𝜑2 and �� = 𝜑1 − 𝜑2 giving the resulting Lagrangian for the two scalar fields𝜑1,2

ℒ𝜒kin = (𝑎1 + 𝑎2)

(𝜑12𝜑1 − 𝜑22𝜑2 −

1

4(𝜑1 − 𝜑2)2

). (2.8)

As a result, the two scalar fields 𝜑1,2 always enter with opposite kinetic terms, signaling that oneof them is always a ghost.3 The only way to prevent this generic pathology is to make the specificchoice 𝑎1 + 𝑎2 = 0, which corresponds to the well-known Maxwell kinetic term.

Helicity-1 mode and gauge symmetry

Now that the form of the local and covariant kinetic term has been uniquely established by therequirement that no ghost rides on top of the helicity-0 mode, we focus on the remaining transversemode 𝐴⊥

𝜇 ,

ℒhelicity−1kin = 𝑎1

(𝜕𝜇𝐴

⊥𝜈

)2, (2.9)

which has the correct normalization if 𝑎1 = −1/2. As a result, the only possible local kinetic termfor a spin-1 field is the Maxwell one:

ℒspin−1kin = −1

4𝐹 2𝜇𝜈 (2.10)

with 𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇. Restricting ourselves to a massless spin-1 field (with no potential andother interactions), the resulting Maxwell theory satisfies the following 𝑈(1) gauge symmetry:

𝐴𝜇 → 𝐴𝜇 + 𝜕𝜇𝜉 . (2.11)

This gauge symmetry projects out two of the naive four degrees of freedom. This can be seen atthe level of the Lagrangian directly, where the gauge symmetry (2.11) allows us to fix the gaugeof our choice. For convenience, we perform a (3 + 1)-split and choose Coulomb gauge 𝜕𝑖𝐴

𝑖 = 0,so that only two dofs are present in 𝐴𝑖, i.e., 𝐴𝑖 contains no longitudinal mode, 𝐴𝑖 = 𝐴𝑡𝑖 + 𝜕𝑖𝐴

𝑙,with 𝜕𝑖𝐴𝑡𝑖 = 0 and the Coulomb gauge sets the longitudinal mode 𝐴𝑙 = 0. The time-component𝐴0 does not exhibit a kinetic term,

ℒspin−1kin =

1

2(𝜕𝑡𝐴𝑖)

2 − 1

2(𝜕𝑖𝐴

0)2 − 1

4(𝜕𝑖𝐴𝑗)

2 , (2.12)

and appears instead as a Lagrange multiplier imposing the constraint

𝜕𝑖𝜕𝑖𝐴0 ≡ 0 . (2.13)

The Maxwell action has therefore only two propagating dofs in 𝐴𝑡𝑖,


2(𝜕𝜇𝐴

𝑡𝑖)

2 . (2.14)

To summarize, the Maxwell kinetic term for a vector field and the fact that a massless vector fieldin four dimensions only propagates 2 dofs is not a choice but has been imposed upon us by therequirement that no ghost rides along with the helicity-0 mode. The resulting theory is enrichedby a 𝑈(1) gauge symmetry which in turn freezes the helicity-0 mode when no mass term is present.We now ‘promote’ the theory to a massive vector field.

2 The equation of motion with respect to 𝜒 gives 2�� = 0, however this should be viewed as a dynamical relationfor ��, which should not be plugged back into the action. On the other hand, when deriving the equation of motionwith respect to ��, we obtain a constraint equation for ��: �� = 22𝜒 which can be plugged back into the action (and𝜒 is then treated as the dynamical field).

3 This is already a problem at the classical level, well before the notion of particle needs to be defined, since clas-sical configurations with arbitrarily large 𝜑1 can always be constructed by compensating with a large configurationfor 𝜑2 at no cost of energy (or classical Hamiltonian).



12 Claudia de Rham

2.1.2 Proca mass term

Starting with the Maxwell action, we consider a covariant mass term 𝐴𝜇𝐴𝜇 corresponding to the

Proca action

ℒProca = −1

4𝐹 2𝜇𝜈 −

1

2𝑚2𝐴𝜇𝐴

𝜇 , (2.15)

and emphasize that the presence of a mass term does not change the fact that the kinetic has beenuniquely fixed by the requirement of the absence of ghost. An immediate consequence of the Procamass term is the breaking of the 𝑈(1) gauge symmetry (2.11), so that the Coulomb gauge can nolonger be chosen and the longitudinal mode is now dynamical. To see this, let us use the previousdecomposition 𝐴𝜇 = 𝐴⊥

𝜇 + 𝜕𝜇�� and notice that the mass term now introduces a kinetic term forthe helicity-0 mode 𝜒 = 𝑚��,

ℒProca = −1

2(𝜕𝜇𝐴

⊥𝜈 )

2 − 1

2𝑚2(𝐴⊥

𝜇 )2 − 1

2(𝜕𝜇𝜒)

2 . (2.16)

A massive vector field thus propagates three dofs, namely two in the transverse modes 𝐴⊥𝜇 and one

in the longitudinal mode 𝜒. Physically, this can be understood by the fact that a massive vectorfield does not propagate along the light-cone, and the fluctuations along the line of propagationcorrespond to an additional physical dof.

Before moving to the Abelian Higgs mechanism, which provides a dynamical way to give amass to bosons, we first comment on the discontinuity in number of dofs between the massiveand massless case. When considering the Proca action (2.16) with the properly normalized fields𝐴⊥𝜇 and 𝜒, one does not recover the massless Maxwell action (2.9) or (2.10) when sending the

boson mass 𝑚 → 0. A priori, this seems to signal the presence of a discontinuity which wouldallow us to distinguish between for instance a massless photon and a massive one no matter howtiny the mass. In practice, however, the difference is physically indistinguishable so long as thephoton couples to external sources in a way which respects the 𝑈(1) symmetry. Note however thatquantum anomalies remain sensitive to the mass of the field so the discontinuity is still present atthis level, see Refs. [197, 204].

To physically tell the difference between a massless vector field and a massive one with tinymass, one has to probe the system, or in other words include interactions with external sources

ℒsources = −𝐴𝜇𝐽𝜇 . (2.17)

The 𝑈(1) symmetry present in the massless case is preserved only if the external sources areconserved, 𝜕𝜇𝐽

𝜇 = 0. Such a source produces a vector field which satisfies

2𝐴⊥𝜇 = 𝐽𝜇 (2.18)

in the massless case. The exchange amplitude between two conserved sources 𝐽𝜇 and 𝐽 ′𝜇 mediated

by a massless vector field is given by

𝒜massless𝐽𝐽 ′ =

∫d4𝐴⊥

𝜇 𝐽′𝜇 =

∫d4𝑥𝐽 ′𝜇 1

2𝐽𝜇 . (2.19)

On the other hand, if the vector field is massive, its response to the source 𝐽𝜇 is instead

(2−𝑚2)𝐴⊥𝜇 = 𝐽𝜇 and 2𝜒 = 0 . (2.20)

In that case, one needs to consider both the transverse and the longitudinal modes of the vectorfield in the exchange amplitude between the two sources 𝐽𝜇 and 𝐽 ′

𝜇. Fortunately, a conserved



Massive Gravity 13

source does not excite the longitudinal mode and the exchange amplitude is uniquely given by thetransverse mode,

𝒜massive𝐽𝐽 ′ =

∫d4𝑥

(𝐴⊥𝜇 + 𝜕𝜇𝜒

)𝐽 ′𝜇 =

∫d4𝑥𝐽 ′𝜇 1

2−𝑚2𝐽𝜇 . (2.21)

As a result, the exchange amplitude between two conserved sources is the same in the limit 𝑚→ 0no matter whether the vector field is intrinsically massive and propagates 3 dofs or if it is masslessand only propagates 2 modes. It is, therefore, impossible to probe the difference between an exactlymassive vector field and a massive one with arbitrarily small mass.

Notice that in the massive case no 𝑈(1) symmetry is present and the source needs not beconserved. However, the previous argument remains unchanged so long as 𝜕𝜇𝐽

𝜇 goes to zero in themassless limit at least as quickly as the mass itself. If this condition is violated, then the helicity-0mode ought to be included in the exchange amplitude (2.21). In parallel, in the massless case thenon-conserved source provides a new kinetic term for the longitudinal mode which then becomesdynamical.

2.1.3 Abelian Higgs mechanism for electromagnetism

Associated with the absence of an intrinsic discontinuity in the massless limit is the existence of aHiggs mechanism for the vector field whereby the vector field acquires a mass dynamically. As weshall see later, the situation is different for gravity where no equivalent dynamical Higgs mechanismhas been discovered to date. Nevertheless, the tools used to describe the Abelian Higgs mechanismand in particular the introduction of a Stuckelberg field will prove useful in the gravitational caseas well.

To describe the Abelian Higgs mechanism, we start with a vector field 𝐴𝜇 with associatedMaxwell tensor 𝐹𝜇𝜈 and a complex scalar field 𝜑 with quartic potential

ℒAH = −1

4𝐹 2𝜇𝜈 −

1

2(𝒟𝜇𝜑) (𝒟𝜇𝜑)* − 𝜆

(𝜑𝜑* − Φ2

0

)2. (2.22)

The covariant derivative, 𝒟𝜇 = 𝜕𝜇 − 𝑖𝑞𝐴𝜇 ensures the existence of the 𝑈(1) symmetry, which inaddition to (2.11) shifts the scalar field as

𝜑→ 𝜑𝑒𝑖𝑞𝜉 . (2.23)

Splitting the complex scalar field 𝜑 into its norm and phase 𝜑 = 𝜙𝑒𝑖𝜒, we see that the covariantderivative plays the role of the mass term for the vector field, when scalar field acquires a non-vanishing vacuum expectation value (vev),

ℒAH = −1

4𝐹 2𝜇𝜈 −

1

2𝜙2 (𝑞𝐴𝜇 − 𝜕𝜇𝜒)2 −

1

2(𝜕𝜇𝜙)

2 − 𝜆(𝜙2 − Φ2

0

)2. (2.24)

The Higgs field 𝜙 can be made arbitrarily massive by setting 𝜆≫ 1 in such a way that its dynamicsmay be neglected and the field can be treated as frozen at 𝜙 ≡ Φ0 = const. The resulting theoryis that of a massive vector field,

ℒAH = −1

4𝐹 2𝜇𝜈 −

1

2Φ2

0 (𝑞𝐴𝜇 − 𝜕𝜇𝜒)2, (2.25)

where the phase 𝜒 of the complex scalar field plays the role of a Stuckelberg which restores the𝑈(1) gauge symmetry in the massive case,

𝐴𝜇 → 𝐴𝜇 + 𝜕𝜇𝜉(𝑥) (2.26)

𝜒→ 𝜒+ 𝑞 𝜉(𝑥) . (2.27)



14 Claudia de Rham

In this formalism, the 𝑈(1) gauge symmetry is restored at the price of introducing explicitly aStuckelberg field which transforms in such a way so as to make the mass term invariant. The sym-metry ensures that the vector field 𝐴𝜇 propagates only 2 dofs, while the Stuckelberg 𝜒 propagatesthe third dof. While no equivalent to the Higgs mechanism exists for gravity, the same Stuckelbergtrick to restore the symmetry can be used in that case. Since the in that context the symmetrybroken is coordinate transformation invariance, (full diffeomorphism invariance or covariance), fourStuckelberg fields should in principle be included in the context of massive gravity, as we shall seebelow.

2.1.4 Interacting spin-1 fields

Now that we have introduced the notion of a massless and a massive spin-1 field, let us look at 𝑁

interacting spin-1 fields. We start with 𝑁 free and massless gauge fields, 𝐴(𝑎)𝜇 , with 𝑎 = 1, · · · , 𝑁 ,

and respective Maxwell tensors 𝐹(𝑎)𝜇𝜈 = 𝜕𝜇𝐴

(𝑎) − 𝜕𝜈𝐴(𝑎)𝜇 ,

ℒN spin−1kin = −1

4

𝑁∑𝑎=1

(𝐹 (𝑎)𝜇𝜈

)2. (2.28)

The theory is then manifestly Abelian and invariant under 𝑁 copies of 𝑈(1), (i.e., the symmetrygroup is 𝑈(1)𝑁 which is Abelian as opposed to 𝑈(𝑁) which would correspond to a Yang–Millstheory and would not be Abelian).

However, in addition to these 𝑁 gauge invariances, the kinetic term is invariant under globalrotations in field space,

𝐴(𝑎)𝜇 −→ 𝐴(𝑎)

𝜇 = 𝑂𝑎𝑏𝐴(𝑏)𝜇 , (2.29)

where 𝑂𝑎𝑏 is a (global) rotation matrix. Now let us consider some interactions between thesedifferent fields. At the linear level (quadratic level in the action), the most general set of interactionsis

ℒint = −1

2

∑𝑎,𝑏

ℐ𝑎𝑏𝐴(𝑎)𝜇 𝐴(𝑏)

𝜈 𝜂𝜇𝜈 , (2.30)

where ℐ𝑎𝑏 is an arbitrary symmetric matrix with constant coefficients. For an arbitrary rank-Nmatrix, all 𝑁 copies of 𝑈(1) are broken, and the theory then propagates 𝑁 additional helicity-0modes, for a total of 3𝑁 independent polarizations in four spacetime dimensions. However, if therank 𝑟 of ℐ is 𝑟 < 𝑁 , i.e., if some of the eigenvalues of ℐ vanish, then there are 𝑁 − 𝑟 specialdirections in field space which receive no interactions, and the theory thus keeps 𝑁−𝑟 independentcopies of 𝑈(1). The theory then propagates 𝑟 massive spin-1 fields and 𝑁−𝑟 massless spin-2 fields,for a total of 3𝑁 − 𝑟 independent polarizations in four dimensions.

We can see this statement more explicitly in the case of 𝑁 spin-1 fields by diagonalizing themass matrix ℐ. A mentioned previously, the kinetic term is invariant under field space rotations,(2.29), so one can use this freedom to work in a field representation where the mass matrix 𝐼 isdiagonal,

ℐ𝑎𝑏 = diag(𝑚2

1, · · · ,𝑚2𝑁

). (2.31)

In this representation the gauge fields are the mass eigenstates and the mass spectrum is simplygiven by the eigenvalues of ℐ𝑎𝑏.

2.2 Spin-2 field

As we have seen in the case of a vector field, as long as it is local and Lorentz-invariant, the kineticterm is uniquely fixed by the requirement that no ghost be present. Moving now to a spin-2 field,the same argument applies exactly and the Einstein–Hilbert term appears naturally as the unique



Massive Gravity 15

kinetic term free of any ghost-like instability. This is possible thanks to a symmetry which projectsout all unwanted dofs, namely diffeomorphism invariance (linear diffs at the linearized level, andnon-linear diffs/general covariance at the non-linear level).

2.2.1 Einstein–Hilbert kinetic term

We consider a symmetric Lorentz tensor field ℎ𝜇𝜈 . The kinetic term can be decomposed into fourpossible local contributions (assuming Lorentz invariance and ignoring terms which are equivalentupon integration by parts):

ℒspin−2kin =

1

2𝜕𝛼ℎ𝜇𝜈

(𝑏1𝜕𝛼ℎ𝜇𝜈 + 2𝑏2𝜕(𝜇ℎ𝜈)𝛼 + 𝑏3𝜕𝛼ℎ𝜂𝜇𝜈 + 2𝑏4𝜕(𝜇ℎ𝜂𝜈)𝛼

), (2.32)

where 𝑏1,2,3,4 are dimensionless coefficients which are to be determined in the same way as for thevector field. We split the 10 components of the symmetric tensor field ℎ𝜇𝜈 into a transverse tensorℎ𝑇𝜇𝜈 (which carries 6 components) and a vector field 𝜒𝜇 (which carries 4 components),

ℎ𝜇𝜈 = ℎ𝑇𝜇𝜈 + 2𝜕(𝜇𝜒𝜈) . (2.33)

Just as in the case of the spin-1 field, an arbitrary kinetic term of the form (2.32) with untunedcoefficients 𝑏𝑖 would contain higher derivatives for 𝜒𝜇 which in turn would imply a ghost. As weshall see below, avoiding a ghost within the kinetic term automatically leads to gauge-invariance.After substitution of ℎ𝜇𝜈 in terms of ℎ𝑇𝜇𝜈 and 𝜒𝜇, the potentially dangerous parts are

ℒspin−2kin ⊃ (𝑏1 + 𝑏2)𝜒

𝜇22𝜒𝜇 + (𝑏1 + 3𝑏2 + 2𝑏3 + 4𝑏4)𝜒𝜇2𝜕𝜇𝜕𝜈𝜒

𝜈 (2.34)

−2ℎ𝑇𝜇𝜈((𝑏2 + 𝑏4)𝜕𝜇𝜕𝜈𝜕𝛼𝜒

𝛼 + (𝑏1 + 𝑏2)𝜕𝜇2𝜒𝜇

+ (𝑏3 + 𝑏4)2𝜕𝛼𝜒𝛼 𝜂𝜇𝜈

).

Preventing these higher derivative terms from arising sets

𝑏4 = −𝑏3 = −𝑏2 = 𝑏1 , (2.35)

or in other words, the unique (local and Lorentz-invariant) kinetic term one can write for a spin-2field is the Einstein–Hilbert term


4ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 = −1

4ℎ𝑇𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝑇𝛼𝛽 , (2.36)

where ℰ is the Lichnerowicz operator

ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 = −1

2

(2ℎ𝜇𝜈 − 2𝜕(𝜇𝜕𝛼ℎ

𝛼𝜈) + 𝜕𝜇𝜕𝜈ℎ− 𝜂𝜇𝜈(2ℎ− 𝜕𝛼𝜕𝛽ℎ𝛼𝛽)

), (2.37)

and we have set 𝑏1 = −1/4 to follow standard conventions. As a result, the kinetic term for thetensor field ℎ𝜇𝜈 is invariant under the following gauge transformation,

ℎ𝜇𝜈 → ℎ𝜇𝜈 + 𝜕(𝜇𝜉𝜈) . (2.38)

We emphasize that the form of the kinetic term and its gauge invariance is independent on whetheror not the tensor field has a mass, (as long as we restrict ourselves to a local and Lorentz-invariantkinetic term). However, just as in the case of a massive vector field, this gauge invariance cannotbe maintained by a mass term or any other self-interacting potential. So only in the massless case,does this symmetry remain exact. Out of the 10 components of a tensor field, the gauge symmetryremoves 2×4 = 8 of them, leaving a massless tensor field with only two propagating dofs as is wellknown from the propagation of gravitational waves in four dimensions.

In 𝑑 ≥ 3 spacetime dimensions, gravitational waves have 𝑑(𝑑+1)/2−2𝑑 = 𝑑(𝑑−3)/2 independentpolarizations. This means that in three dimensions there are no gravitational waves and in fivedimensions they have five independent polarizations.



16 Claudia de Rham

2.2.2 Fierz–Pauli mass term

As seen in seen in Section 2.2.1, for a local and Lorentz-invariant theory, the linearized kineticterm is uniquely fixed by the requirement that longitudinal modes propagate no ghost, which inturn prevents that operator from exciting these modes altogether. Just as in the case of a massivespin-1 field, we shall see in what follows that the longitudinal modes can nevertheless be excitedwhen including a mass term. In what follows we restrict ourselves to linear considerations andspare any non-linearity discussions for Parts I and II. See also [327] for an analysis of the linearizedFierz–Pauli theory using Bardeen variables.

In the case of a spin-2 field ℎ𝜇𝜈 , we are a priori free to choose between two possible mass termsℎ2𝜇𝜈 and ℎ2, so that the generic mass term can be written as a combination of both,

ℒmass = −1

8𝑚2(ℎ2𝜇𝜈 −𝐴ℎ2

), (2.39)

where 𝐴 is a dimensionless parameter. Just as in the case of the kinetic term, the stability of thetheory constrains very strongly the phase space and we shall see that only for 𝛼 = 1 is the theorystable at that order. The presence of this mass term breaks diffeomorphism invariance. Restoringit requires the introduction of four Stuckelberg fields 𝜒𝜇 which transform under linear diffeomor-phisms in such a way as to make the mass term invariant, just as in the Abelian-Higgs mechanismfor electromagnetism [174]. Including the four linearized Stuckelberg fields, the resulting massterm

ℒmass = −1

8𝑚2((ℎ𝜇𝜈 + 2𝜕(𝜇𝜒𝜈))

2 −𝐴(ℎ+ 2𝜕𝛼𝜒𝛼)2), (2.40)

is invariant under the simultaneous transformations:

ℎ𝜇𝜈 → ℎ𝜇𝜈 + 𝜕(𝜇𝜉𝜈) , (2.41)

𝜒𝜇 → 𝜒𝜇 −1

2𝜉𝜇 . (2.42)

This mass term then provides a kinetic term for the Stuckelberg fields

ℒ𝜒kin = −1

2𝑚2((𝜕𝜇𝜒𝜈)

2 −𝐴(𝜕𝛼𝜒𝛼)2), (2.43)

which is precisely of the same form as the kinetic term considered for a spin-1 field (2.1) inSection 2.1.1 with 𝑎3 = 0 and 𝑎2 = 𝐴𝑎1. Now the same logic as in Section 2.1.1 applies and singlingout the longitudinal component of these Stuckelberg fields it follows that the only combinationwhich does not involve higher derivatives is 𝑎2 = 𝑎1 or in other words 𝐴 = 1. As a result, theonly possible mass term one can consider which is free from an Ostrogradsky instability is theFierz–Pauli mass term

ℒFPmass = −1

8𝑚2((ℎ𝜇𝜈 + 2𝜕(𝜇𝜒𝜈))

2 − (ℎ+ 2𝜕𝛼𝜒𝛼)2). (2.44)

In unitary gauge, i.e., in the gauge where the Stuckelberg fields 𝜒𝑎 are set to zero, the Fierz–Paulimass term simply reduces to

ℒFPmass = −1

8𝑚2(ℎ2𝜇𝜈 − ℎ2

), (2.45)

where once again the indices are raised and lowered with respect to the Minkowski metric.



Massive Gravity 17

Propagating degrees of freedom

To identify the propagating degrees of freedom we may split 𝜒𝑎 further into a transverse and alongitudinal mode,

𝜒𝑎 =1

𝑚𝐴𝑎 +

1

𝑚2𝜂𝑎𝑏𝜕𝑏𝜋 , (2.46)

(where the normalization with negative factors of 𝑚 has been introduced for further convenience).In terms of ℎ𝜇𝜈 and the Stuckelberg fields 𝐴𝜇 and 𝜋 the linearized Fierz–Pauli action is

ℒFP = −1

4ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 −

1

2ℎ𝜇𝜈 (Π𝜇𝜈 − [Π]𝜂𝜇𝜈)−

1

8𝐹 2𝜇𝜈 (2.47)

−1

8𝑚2(ℎ2𝜇𝜈 − ℎ2

)− 1

2𝑚 (ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈) 𝜕(𝜇𝐴𝜈) ,

with 𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 −𝜕𝜈𝐴𝜇 and Π𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋 and all the indices are raised and lowered with respectto the Minkowski metric.

Terms on the first line represent the kinetic terms for the different fields while the second linerepresent the mass terms and mixing.

We see that the kinetic term for the field 𝜋 is hidden in the mixing with ℎ𝜇𝜈 . To make the field

content explicit, we may diagonalize this mixing by shifting ℎ𝜇𝜈 = ℎ𝜇𝜈 + 𝜋𝜂𝜇𝜈 and the linearizedFierz–Pauli action is

ℒFP = −1


3

4(𝜕𝜋)2 − 1

8𝐹 2𝜇𝜈 (2.48)

−1

8𝑚2(ℎ2𝜇𝜈 − ℎ2) +

3

2𝑚2𝜋2 +

3

2𝑚2𝜋ℎ

−1

2𝑚(ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈)𝜕(𝜇𝐴𝜈) + 3𝑚𝜋𝜕𝛼𝐴

𝛼 .

This decomposition allows us to identify the different degrees of freedom present in massive gravity(at least at the linear level): ℎ𝜇𝜈 represents the helicity-2 mode as already present in GR andpropagates 2 dofs, 𝐴𝜇 represents the helicity-1 mode and propagates 2 dofs, and finally 𝜋 representsthe helicity-0 mode and propagates 1 dof, leading to a total of five dofs as is to be expected for amassive spin-2 field in four dimensions.

The degrees of freedom have not yet been split into their mass eigenstates but on doing so onecan easily check that all the degrees of freedom have the same positive mass square 𝑚2.

Most of the phenomenology and theoretical consistency of massive gravity is related to thedynamics of the helicity-0 mode. The coupling to matter occurs via the coupling ℎ𝜇𝜈𝑇

𝜇𝜈 =

ℎ𝜇𝜈𝑇𝜇𝜈 + 𝜋𝑇 , where 𝑇 is the trace of the external stress-energy tensor. We see that the helicity-0

mode couples directly to conserved sources (unlike in the case of the Proca field) but the helicity-1mode does not. In most of what follows we will thus be able to ignore the helicity-1 mode.

Higgs mechanism for gravity

As we shall see in Section 9.1, the graviton mass can also be promoted to a scalar function of one ormany other fields (for instance of a different scalar field), 𝑚 = 𝑚(𝜓). We can thus wonder whethera dynamical Higgs mechanism for gravity can be considered where the field(s) 𝜓 start in a phase forwhich the graviton mass vanishes, 𝑚(𝜓) = 0 and dynamically evolves to acquire a non-vanishingvev for which 𝑚(𝜓) = 0. Following the same logic as the Abelian Higgs for electromagnetism, thisstrategy can only work if the number of dofs in the massless phase 𝑚 = 0 is the same as that inthe massive case 𝑚 = 0. Simply promoting the mass to a function of an external field is thus notsufficient since the graviton helicity-0 and -1 modes would otherwise be infinitely strongly coupledas 𝑚→ 0.



18 Claudia de Rham

To date no candidate has been proposed for which the graviton mass could dynamically evolvefrom a vanishing value to a finite one without falling into such strong coupling issues. This doesnot imply that Higgs mechanism for gravity does not exist, but as yet has not been found. Forinstance on AdS, there could be a Higgs mechanism as proposed in [431], where the mass term comesfrom integrating out some conformal fields with slightly unusual (but not unphysical) ‘transparent’boundary conditions. This mechanism is specific to AdS and to the existence of time-like boundaryand would not apply on Minkowski or dS.

2.2.3 Van Dam–Veltman–Zakharov discontinuity

As in the case of spin-1, the massive spin-2 field propagates more dofs than the massless one.Nevertheless, these new excitations bear no observational signatures for the spin-1 field whenconsidering an arbitrarily small mass, as seen in Section 2.1.2. The main reason for that is thatthe helicity-0 polarization of the photon couple only to the divergence of external sources whichvanishes for conserved sources. As a result no external sources directly excite the helicity-0 modeof a massive spin-1 field. For the spin-2 field, on the other hand, the situation is different as thehelicity-0 mode can now couple to the trace of the stress-energy tensor and so generic sources willexcite not only the 2 helicity-2 polarization of the graviton but also a third helicity-0 polarization,which could in principle have dramatic consequences. To see this more explicitly, let us computethe gravitational exchange amplitude between two sources 𝑇𝜇𝜈 and 𝑇 ′𝜇𝜈 in both the massive andmassless gravitational cases.

In the massless case, the theory is diffeomorphism invariant. When considering coupling toexternal sources, of the form ℎ𝜇𝜈𝑇

𝜇𝜈 , we thus need to ensure that the symmetry be preserved,which implies that the stress-energy tensor 𝑇𝜇𝜈 should be conserved 𝜕𝜇𝑇

𝜇𝜈 = 0. When computingthe gravitational exchange amplitude between two sources we thus restrict ourselves to conservedones. In the massive case, there is a priori no reasons to restrict ourselves to conserved sources, solong as their divergences cancel in the massless limit 𝑚→ 0.

Massive spin-2 field

Let us start with the massive case, and consider the response to a conserved external source 𝑇𝜇𝜈 ,

ℒ = −1


𝑚2

8(ℎ2𝜇𝜈 − ℎ2) +

1

2𝑀Plℎ𝜇𝜈𝑇

𝜇𝜈 . (2.49)

The linearized Einstein equation is then

ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 +1

2𝑚2(ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈) =

1

𝑀Pl𝑇𝜇𝜈 . (2.50)

To solve this modified linearized Einstein equation for ℎ𝜇𝜈 we consider the trace and the divergenceseparately,

ℎ = − 1

3𝑚2𝑀Pl

(𝑇 +

2

𝑚2𝜕𝛼𝜕𝛽𝑇

𝛼𝛽

)(2.51)

𝜕𝜇ℎ𝜇𝜈 =

1

𝑚2𝑀Pl

(𝜕𝜇𝑇

𝜇𝜈 +

1

3𝜕𝜈𝑇 +

2

3𝑚2𝜕𝜈𝜕𝛼𝜕𝛽𝑇

𝛼𝛽

). (2.52)

As is already apparent at this level, the massless limit𝑚→ 0 is not smooth which is at the origin ofthe vDVZ discontinuity (for instance we see immediately that for a conserved source the linearizedRicci scalar vanishes 𝜕𝜇𝜕𝜈ℎ

𝜇𝜈 − 2ℎ = 0 see Refs. [465, 497]. This linearized vDVZ discontinuitywas recently repointed out in [193].) As has been known for many decades, this discontinuity (or



Massive Gravity 19

the fact that the Ricci scalar vanishes) is an artefact of the linearized theory and is resolved bythe Vainshtein mechanism [463] as we shall see later.

Plugging these expressions back into the modified Einstein equation, we get(2−𝑚2

)ℎ𝜇𝜈 = − 1

𝑀Pl

[𝑇𝜇𝜈 −

1

3𝑇𝜂𝜇𝜈 −

2

𝑚2𝜕(𝜇𝜕𝛼𝑇

𝛼𝜈) +

1

3𝑚2𝜕𝜇𝜕𝜈𝑇 (2.53)

+1

3𝑚2𝜕𝛼𝜕𝛽𝑇

𝛼𝛽𝜂𝜇𝜈 +2

3𝑚4𝜕𝜇𝜕𝜈𝜕𝛼𝜕𝛽𝑇

𝛼𝛽]

=1

𝑀Pl

[𝜂𝜇(𝛼𝜂𝜈𝛽) −

1

3𝜂𝜇𝜈𝜂𝛼𝛽

]𝑇𝛼𝛽 , (2.54)

with

𝜂𝜇𝜈 = 𝜂𝜇𝜈 −1

𝑚2𝜕𝜇𝜕𝜈 . (2.55)

The propagator for a massive spin-2 field is thus given by

𝐺massive𝜇𝜈𝛼𝛽 (𝑥, 𝑥′) =

𝑓massive𝜇𝜈𝛼𝛽

2−𝑚2, (2.56)

where 𝑓massive𝜇𝜈𝛼𝛽 is the polarization tensor,

𝑓massive𝜇𝜈𝛼𝛽 = 𝜂𝜇(𝛼𝜂𝜈𝛽) −

1

3𝜂𝜇𝜈𝜂𝛼𝛽 . (2.57)

In Fourier space we have

𝑓massive𝜇𝜈𝛼𝛽 (𝑝𝜇,𝑚) =

2

3𝑚4𝑝𝜇𝑝𝜈𝑝𝛼𝑝𝛽 + 𝜂𝜇(𝛼𝜂𝜈𝛽) −

1

3𝜂𝜇𝜈𝜂𝛼𝛽 (2.58)

+1

𝑚2

(𝑝𝛼𝑝(𝜇𝜂𝜈)𝛽 + 𝑝𝛽𝑝(𝜇𝜂𝜈)𝛼 −

1

3𝑝𝜇𝑝𝜈𝜂𝛼𝛽 −

1

3𝑝𝛼𝑝𝛽𝜂𝜇𝜈

).

The amplitude exchanged between two sources 𝑇𝜇𝜈 and 𝑇 ′𝜇𝜈 via a massive spin-2 field is thus given

by

𝒜massive𝑇𝑇 ′ =

∫d4𝑥 ℎ𝜇𝜈𝑇

′𝜇𝜈 =

∫d4𝑥 𝑇 ′𝜇𝜈 𝑓

massive𝜇𝜈𝛼𝛽

2−𝑚2𝑇𝛼𝛽 . (2.59)

As mentioned previously, to compare this result with the massless case, the sources ought to beconserved in the massless limit, 𝜕𝜇𝑇

𝜇𝜈 , 𝜕𝜇𝑇

𝜇𝜈′ → 0 as𝑚→ 0. The gravitational exchange amplitude

in the massless limit is thus given by

𝒜𝑚→0𝑇𝑇 ′

∫d4𝑥 𝑇 ′𝜇𝜈 1

2

(𝑇𝜇𝜈 −

1

3𝑇𝜂𝜇𝜈

). (2.60)

We now compare this result with the amplitude exchanged by a purely massless graviton.

Massless spin-2 field

In the massless case, the equation of motion (2.50) reduces to the linearized Einstein equation

ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 =1

𝑀Pl𝑇𝜇𝜈 , (2.61)

where diffeomorphism invariance requires the stress-energy to be conserved, 𝜕𝜇𝑇𝜇𝜈 = 0. In this case

the transverse part of this equation is trivially satisfied (as a consequence of the Bianchi identity



20 Claudia de Rham

which follows from symmetry). Since the theory is invariant under diffeomorphism transforma-tions (2.38), one can choose a gauge of our choice, for instance de Donder (or harmonic) gauge

𝜕𝜇ℎ𝜇𝜈 =

1

2𝑝𝜈 . (2.62)

In de Donder gauge, the Einstein equation then reduces to

(2−𝑚2)ℎ𝜇𝜈 = − 2

𝑀Pl

(𝑇𝜇𝜈 −

1

2𝑇𝜂𝜇𝜈

). (2.63)

The propagator for a massless spin-2 field is thus given by

𝐺massless𝜇𝜈𝛼𝛽 =

𝑓massless𝜇𝜈𝛼𝛽

2, (2.64)

where 𝑓massless𝜇𝜈𝛼𝛽 is the polarization tensor,

𝑓massless𝜇𝜈𝛼𝛽 = 𝜂𝜇(𝛼𝜂𝜈𝛽) −

1

2𝜂𝜇𝜈𝜂𝛼𝛽 . (2.65)

The amplitude exchanged between two sources 𝑇𝜇𝜈 and 𝑇 ′𝜇𝜈 via a genuinely massless spin-2 field

is thus given by

𝒜massless𝑇𝑇 ′ = − 2

𝑀Pl

∫d4𝑥 𝑇 ′𝜇𝜈 1

2

(𝑇𝜇𝜈 −

1

2𝑇𝜂𝜇𝜈

), (2.66)

and differs from the result (2.60) in the small mass limit. This difference between the massless limitof the massive propagator and the massless propagator (and gravitational exchange amplitude) isa well-known fact and was first pointed out by van Dam, Veltman and Zakharov in 1970 [465, 497].The resolution to this ‘problem’ lies within the Vainshtein mechanism [463]. In 1972, Vainshteinshowed that a theory of massive gravity becomes strongly coupled a low energy scale when thegraviton mass is small. As a result, the linear theory is no longer appropriate to describe the theoryin the limit of small mass and one should keep track of the non-linear interactions (very much aswhat we do when approaching the Schwarzschild radius in GR.) We shall see in Section 10.1 howa special set of interactions dominate in the massless limit and are responsible for the screening ofthe extra degrees of freedom present in massive gravity.

Another ‘non-GR’ effect was also recently pointed out in Ref. [280] where a linear analysisshowed that massive gravity predicts different spin-orientations for spinning objects.

2.3 From linearized diffeomorphism to full diffeomorphism invariance

When considering the massless and non-interactive spin-2 field in Section 2.2.1, the linear gaugeinvariance (2.38) is exact. However, if this field is to be probed and communicates with the rest ofthe world, the gauge symmetry is forced to include non-linear terms which in turn forces the kineticterm to become fully non-linear. The result is the well-known fully covariant Einstein–Hilbert term𝑀2

Pl

√−𝑔𝑅, where 𝑅 is the scalar curvature associated with the metric 𝑔𝜇𝜈 = 𝜂𝜇𝜈 + ℎ𝜇𝜈/𝑀Pl.

To see this explicitly, let us start with the linearized theory and couple it to an external source𝑇𝜇𝜈0 , via the coupling

ℒlinearmatter =

1


𝜇𝜈0 . (2.67)

This coupling preserves diffeomorphism invariance if the source is conserved, 𝜕𝜇𝑇𝜇𝜈0 = 0. To be

more explicit, let us consider a massless scalar field 𝜙 which satisfies the Klein–Gordon equation2𝜙 = 0. A natural choice for the stress-energy tensor 𝑇𝜇𝜈 is then

𝑇𝜇𝜈0 = 𝜕𝜇𝜙𝜕𝜈𝜙− 1

2(𝜕𝜙)2𝜂𝜇𝜈 , (2.68)



Massive Gravity 21

so that the Klein–Gordon equation automatically guarantees the conservation of the stress-energytensor on-shell at the linear level and linearized diffeomorphism invariance. However, the verycoupling between the scalar field and the spin-2 field affects the Klein–Gordon equation in sucha way that beyond the linear order, the stress-energy tensor given in (2.68) fails to be conserved.When considering the coupling (2.67), the Klein–Gordon equation receives corrections of the orderof ℎ𝜇𝜈/𝑀Pl

2𝜙 =1

𝑀Pl

(𝜕𝛼(ℎ𝛼𝛽𝜕

𝛽𝜙)− 1

2𝜕𝛼(ℎ

𝛽𝛽𝜕

𝛼𝜙)

), (2.69)

implying a failure of conservation of 𝑇𝜇𝜈0 at the same order,

𝜕𝜇𝑇𝜇𝜈0 =

𝜕𝜈𝜙

𝑀Pl

(𝜕𝛼(ℎ𝛼𝛽𝜕

𝛽𝜙)− 1

2𝜕𝛼(ℎ

𝛽𝛽𝜕

𝛼𝜙)

). (2.70)

The resolution is of course to include non-linear corrections in ℎ/𝑀Pl in the coupling with externalmatter,

ℒmatter =1


𝜇𝜈0 +

1

2𝑀2Pl

ℎ𝜇𝜈ℎ𝛼𝛽𝑇𝜇𝜈𝛼𝛽1 + · · · , (2.71)

and promote diffeomorphism invariance to a non-linearly realized gauge symmetry, symbolically,

ℎ→ ℎ+ 𝜕𝜉 +1

𝑀Pl𝜕(ℎ𝜉) + · · · , (2.72)

so this gauge invariance is automatically satisfied on-shell order by order in ℎ/𝑀Pl, i.e., the scalarfield (or general matter field) equations of motion automatically imply the appropriate relation forthe stress-energy tensor to all orders in ℎ/𝑀Pl. The resulting symmetry is the well-known fullynon-linear coordinate transformation invariance (or full diffeomorphism invariance or covariance4),which requires the stress-energy tensor to be covariantly conserved. To satisfy this symmetry, thekinetic term (2.36) should then be promoted to a fully non-linear contribution,

ℒspin−2kin linear = −

1

4ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 −→ ℒspin−2

kin covariant =𝑀2

Pl

2

√−𝑔𝑅[𝑔] . (2.73)

Just as the linearized version ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 was unique, the non-linear realization√−𝑔𝑅 is also

unique.5 As a result, any theory of an interacting spin-2 field is necessarily fully non-linear andleads to the theory of gravity where non-linear diffeomorphism invariance (or covariance) plays therole of the local gauge symmetry that projects out four out of the potential six degrees of freedomof the graviton and prevents the excitation of any ghost by the kinetic term.

The situation is very different from that of a spin-1 field as seen earlier, where coupling withother fields can be implemented at the linear order without affecting the 𝑈(1) gauge symmetry.The difference is that in the case of a 𝑈(1) symmetry, there is a unique nonlinear completion ofthat symmetry, i.e., the unique nonlinear completion of a 𝑈(1) is nothing else but a 𝑈(1). Thusany nonlinear Lagrangian which preserves the full 𝑈(1) symmetry will be a consistent interactingtheory. On the other hand, for spin-2 fields, there are two, and only two ways to nonlinearlycomplete linear diffs, one as linear diffs in the full theory and the other as full non-linear diffs.While it is possible to write self-interactions which preserve linear diffs, there are no interactionsbetween matter and ℎ𝜇𝜈 which preserve linear diffs. Thus any theory of gravity must exhibit fullnonlinear diffs and is in this sense what leads us to GR.

4 In this review, the notion of fully non-linear coordinate transformation invariance is equivalent to that of fulldiffeomorphism invariance or covariance.

5 Up to other Lovelock invariants. Note however that 𝑓(𝑅) theories are not exceptions, as the kinetic term forthe spin-2 field is still given by

√−𝑔𝑅. See Section 5.6 for more a more detailed discussion in the case of massive

gravity.



22 Claudia de Rham

2.4 Non-linear Stuckelberg decomposition

On the need for a reference metric

We have introduced the spin-2 field ℎ𝜇𝜈 as the perturbation about flat spacetime. When consideringthe theory of a field of given spin it is only natural to work with Minkowski as our spacetime metric,since the notion of spin follows from that of Poincare invariance. Now when extending the theorynon-linearly, we may also extend the theory about different reference metric. When dealing witha reference metric different than Minkowski, one loses the interpretation of the field as massivespin-2, but one can still get a consistent theory. One could also wonder whether it is possible towrite a theory of massive gravity without the use of a reference metric at all. This interestingquestion was investigated in [75], where it shown that the only consistent alternative is to considera function of the metric determinant. However, as shown in [75], the consistent function of thedeterminant is the cosmological constant and does not provide a mass for the graviton.

Non-linear Stuckelberg

Full diffeomorphism invariance (or covariance) indicates that the theory should be built out of scalarobjects constructed out of the metric 𝑔𝜇𝜈 and other tensors. However, as explained previously atheory of massive gravity requires the notion of a reference metric6 𝑓𝜇𝜈 (which may be Minkowski𝑓𝜇𝜈 = 𝜂𝜇𝜈) and at the linearized level, the mass for gravity was not built out of the full metric𝑔𝜇𝜈 , but rather out of the fluctuation ℎ𝜇𝜈 about this reference metric which does not transform asa tensor under general coordinate transformations. As a result the mass term breaks covariance.

This result is already transparent at the linear level where the mass term (2.39) breaks linearizeddiffeomorphism invariance. Nevertheless, that gauge symmetry can always be ‘formally’ restoredusing the Stuckelberg trick which amounts to replacing the reference metric (so far we have beenworking with the flat Minkowski metric as the reference), to

𝜂𝜇𝜈 −→ (𝜂𝜇𝜈 −2

𝑀Pl𝜕(𝜇𝜒𝜈)) , (2.74)

and transforming 𝜒𝜇 under linearized diffeomorphism in such a way that the combination ℎ𝜇𝜈 −2𝜕(𝜇𝜒𝜈) remains invariant. Now that the symmetry is non-linearly realized and replaced by generalcovariance, this Stuckelberg trick should also be promoted to a fully covariant realization.

Following the same Stuckelberg trick non-linearly, one can ‘formally restore’ covariance byincluding four Stuckelberg fields 𝜑𝑎 (𝑎 = 0, 1, 2, 3) and promoting the reference metric 𝑓𝜇𝜈 , which

may of may not be Minkowski, to a tensor 𝑓𝜇𝜈 [446, 27],

𝑓𝜇𝜈 −→ 𝑓𝜇𝜈 = 𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑏𝑓𝑎𝑏 (2.75)

As we can see from this last expression, 𝑓𝜇𝜈 transforms as a tensor under coordinate transformationsas long as each of the four fields 𝜑𝑎 transform as scalars. We may now construct the theory ofmassive gravity as a scalar Lagrangian of the tensors 𝑓𝜇𝜈 and 𝑔𝜇𝜈 . In unitary gauge, where the

Stuckelberg fields are 𝜑𝑎 = 𝑥𝑎, we simply recover 𝑓𝜇𝜈 = 𝑓𝜇𝜈 .This Stuckelberg trick for massive gravity dates already from Green and Thorn [267] and

from Siegel [446], introduced then within the context of open string theory. In the same way asthe massless graviton naturally emerges in the closed string sector, open strings also have spin-2 excitations but whose lowest energy state is massive at tree level (they only become massless

6 Strictly speaking, the notion of spin is only meaningful as a representation of the Lorentz group, thus the theoryof massive spin-2 field is only meaningful when Lorentz invariance is preserved, i.e., when the reference metric isMinkowski. While the notion of spin can be extended to other maximally symmetric spacetimes such as AdS anddS, it loses its meaning for non-maximally symmetric reference metrics 𝑓𝜇𝜈 .



Massive Gravity 23

once quantum corrections are considered). Thus at the classical level, open strings contain adescription of massive excitations of a spin-2 field, where gauge invariance is restored thanks tosame Stuckelberg fields as introduced in this section. In open string theory, these Stuckelberg fieldsnaturally arise from the ghost coordinates. When constructing the non-linear theory of massivegravity from extra dimension, we shall see that in that context the Stuckelberg fields naturallyarise at the shift from the extra dimension.

For later convenience, it will be useful to construct the following tensor quantity,

X𝜇𝜈 = 𝑔𝜇𝛼𝑓𝛼𝜈 = 𝜕𝜇𝜑𝑎𝜕𝜈𝜑𝑏𝑓𝑎𝑏 , (2.76)

in unitary gauge, X = 𝑔−1𝑓 .

Alternative Stuckelberg trick

An alternative way to Stuckelberize the reference metric 𝑓𝜇𝜈 is to express it as

𝑔𝑎𝑐𝑓𝑐𝑏 → Y𝑎𝑏 = 𝑔𝜇𝜈𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑐𝑓𝑐𝑏 . (2.77)

As nicely explained in Ref. [14], both matrices 𝑋𝜇𝜈 and 𝑌 𝑎𝑏 have the same eigenvalues, so one

can choose either one of them in the definition of the massive gravity Lagrangian without anydistinction. The formulation in terms of 𝑌 rather than 𝑋 was originally used in Ref. [94], althoughunsuccessfully as the potential proposed there exhibits the BD ghost instability, (see for instanceRef. [60]).

Helicity decomposition

If we now focus on the flat reference metric, 𝑓𝜇𝜈 = 𝜂𝜇𝜈 , we may further split the Stuckelberg fieldsas 𝜑𝑎 = 𝑥𝑎 − 1

𝑀Pl𝜒𝑎 and identify the index 𝑎 with a Lorentz index,7 we obtain the non-linear

generalization of the Stuckelberg trick used in Section 2.2.2

𝜂𝜇𝜈 −→ 𝑓𝜇𝜈 = 𝜂𝜇𝜈 −2

𝑀Pl𝜕(𝜇𝜒𝜈) +

1

𝑀2Pl

𝜕𝜇𝜒𝑎𝜕𝜈𝜒

𝑏𝜂𝑎𝑏 (2.78)

= 𝜂𝜇𝜈 −2

𝑀Pl𝑚𝜕(𝜇𝐴𝜈) −

2

𝑀Pl𝑚2Π𝜇𝜈 (2.79)

+1

𝑀2Pl𝑚

2𝜕𝜇𝐴

𝛼𝜕𝜈𝒜𝛼 +2

𝑀2Pl𝑚

3𝜕𝜇𝐴

𝛼Π𝜈𝛼 +1

𝑀2Pl𝑚

4Π2𝜇𝜈 ,

where in the second equality we have used the split performed in (2.46) of 𝜒𝑎 in terms of thehelicity-0 and -1 modes and all indices are raised and lowered with respect to 𝜂𝜇𝜈 .

In other words, the fluctuations about flat spacetime are promoted to the tensor 𝐻𝜇𝜈

ℎ𝜇𝜈 =𝑀Pl (𝑔𝜇𝜈 − 𝜂𝜇𝜈) −→ 𝐻𝜇𝜈 =𝑀Pl

(𝑔𝜇𝜈 − 𝑓𝜇𝜈

)(2.80)

with

𝐻𝜇𝜈 = ℎ𝜇𝜈 + 2𝜕(𝜇𝜒𝜈) −1

𝑀Pl𝜂𝑎𝑏𝜕𝜇𝜒

𝑎𝜕𝜈𝜒𝑏 (2.81)

= ℎ𝜇𝜈 +2

𝑚𝜕(𝜇𝐴𝜈) +

2

𝑚2Π𝜇𝜈 (2.82)

− 1

𝑀Pl𝑚2𝜕𝜇𝐴

𝛼𝜕𝜈𝒜𝛼 −2

𝑀Pl𝑚3𝜕𝜇𝐴

𝛼Π𝜈𝛼 −1

𝑀Pl𝑚4Π2𝜇𝜈 .

7 This procedure can of course be used for any reference metric, but it fails in identifying the proper physicaldegrees of freedom when dealing with a general reference metric. See Refs. [145, 154] as well as Section 8.3.5 forfurther discussions on that point.



24 Claudia de Rham

The field 𝜒𝑎 are introduced to restore the gauge invariance (full diffeomorphism invariance). Wecan now always set a gauge where ℎ𝜇𝜈 is transverse and traceless at the linearized level and 𝐴𝜇is transverse. In this gauge the quantities ℎ𝜇𝜈 , 𝐴𝜇 and 𝜋 represent the helicity decomposition ofthe metric. ℎ𝜇𝜈 is the helicity-2 part of the graviton, 𝐴𝜇 the helicity-1 part and 𝜋 the helicity-0part. The fact that these quantities continue to correctly identify the physical degrees of freedomnon-linearly in the limit 𝑀Pl →∞ is non-trivial and has been derived in [143].

Non-linear Fierz–Pauli

The most straightforward non-linear extension of the Fierz–Pauli mass term is as follows

ℒ(nl1)FP = −𝑚2𝑀2

Pl

√−𝑔([(I− X)2]− [I− X]2

), (2.83)

this mass term is then invariant under non-linear coordinate transformations. This non-linearformulation was used for instance in [27]. Alternatively, one may also generalize the Fierz–Paulimass non-linearly as follows [75]

ℒ(nl2)FP = −𝑚2𝑀2

Pl

√−𝑔√detX

([(I− X−1)2]− [I− X−1]2

). (2.84)

A priori, the linear Fierz–Pauli action for massive gravity can be extended non-linearly in anarbitrary number of ways. However, as we shall see below, most of these generalizations generatea ghost non-linearly, known as the Boulware–Deser (BD) ghost. In Part II, we shall see that theextension of the Fierz–Pauli to a non-linear theory free of the BD ghost is unique (up to twoconstant parameters).

2.5 Boulware–Deser ghost

The easiest way to see the appearance of a ghost at the non-linear level is to follow the Stuckelbergtrick non-linearly and observe the appearance of an Ostrogradsky instability [111, 173], althoughthe original formulation was performed in unitary gauge in [75] in the ADM language (Arnowitt,Deser and Misner, see Ref. [29]). In this section we shall focus on the flat reference metric,𝑓𝜇𝜈 = 𝜂𝜇𝜈 .

Focusing solely on the helicity-0 mode 𝜋 to start with, the tensor X𝜇𝜈 defined in (2.76) isexpressed as

X𝜇𝜈 = 𝛿𝜇𝜈 −2

𝑀Pl𝑚2Π𝜇𝜈 +

1

𝑀2Pl𝑚

4Π𝜇𝛼Π

𝛼𝜈 , (2.85)

where at this level all indices are raised and lowered with respect to the flat reference metric 𝜂𝜇𝜈 .Then the Fierz–Pauli mass term (2.83) reads

ℒ(nl1)FP, 𝜋 = − 4

𝑚2

([Π2]− [Π]2

)+

4

𝑀Pl𝑚4

([Π3]− [Π][Π2]

)+

1

𝑀2Pl𝑚

6

([Π4]− [Π2]2

). (2.86)

Upon integration by parts, we notice that the quadratic term in (2.86) is a total derivative, whichis another way to see the special structure of the Fierz–Pauli mass term. Unfortunately this specialfact does not propagate to higher order and the cubic and quartic interactions are genuine higherorder operators which lead to equations of motion with quartic and cubic derivatives. In otherwords these higher order operators

([Π3]− [Π][Π2]

)and

([Π4]− [Π2]2

)propagate an additional

degree of freedom which by Ostrogradsky’s theorem, always enters as a ghost. While at the linearlevel, these operators might be irrelevant, their existence implies that one can always find anappropriate background configuration 𝜋 = 𝜋0 + 𝛿𝜋, such that the ghost is manifest

ℒ(nl1)FP, 𝜋 =

4

𝑀Pl𝑚4𝑍𝜇𝜈𝛼𝛽𝜕𝜇𝜕𝜈𝛿𝜋𝜕𝛼𝜕𝛽𝛿𝜋 , (2.87)



Massive Gravity 25

with 𝑍𝜇𝜈𝛼𝛽 = 3𝜕𝜇𝜕𝛼𝜋0𝜂𝜈𝛽 − 2𝜋0𝜂

𝜇𝛼𝜂𝜈𝛽 − 2𝜕𝜇𝜕𝜈𝜋0𝜂𝛼𝛽 + · · · . This implies that non-linearly (or

around a non-trivial background), the Fierz–Pauli mass term propagates an additional degree offreedom which is a ghost, namely the BD ghost. The mass of this ghost depends on the backgroundconfiguration 𝜋0,

𝑚2ghost ∼

𝑀Pl𝑚4

𝜕2𝜋0. (2.88)

As we shall see below, the resolution of the vDVZ discontinuity lies in the Vainshtein mechanismfor which the field takes a large vacuum expectation value, 𝜕2𝜋0 ≫ 𝑀Pl𝑚

2, which in the presentcontext would lead to a ghost with an extremely low mass, 𝑚2

ghost . 𝑚2.Choosing another non-linear extension for the Fierz–Pauli mass term as in (2.84) does not seem

to help much,

ℒ(nl2)FP, 𝜋 = − 4

𝑚2

([Π2]− [Π]2

)− 4

𝑀Pl𝑚4

([Π]3 − 4[Π][Π2] + 3[Π3]

)+ · · ·

→ 4

𝑀Pl𝑚4

([Π][Π2]− [Π3]

)+ · · · (2.89)

where we have integrated by parts on the second line, and we recover exactly the same type ofhigher derivatives already at the cubic level, so the BD ghost is also present in (2.84).

Alternatively the mass term was also generalized to include curvature invariants as in Ref. [69].This theory was shown to be ghost-free at the linear level on FLRW but not yet non-linearly.

Function of the Fierz–Pauli mass term

As an extension of the Fierz–Pauli mass term, one could instead write a more general function ofit, as considered in Ref. [75]

ℒ𝐹 (FP) = −𝑚2√−𝑔𝐹(𝑔𝜇𝜈𝑔𝛼𝛽(𝐻𝜇𝛼𝐻𝜈𝛽 −𝐻𝜇𝜈𝐻𝛼𝛽)

), (2.90)

however, one can easily see, if a mass term is actually present, i.e., 𝐹 ′ = 0, there is no analyticchoice of the function 𝐹 which would circumvent the non-linear propagation of the BD ghost.Expanding 𝐹 into a Taylor expansion, we see for instance that the only choice to prevent the cubichigher-derivative interactions in 𝜋, [Π3]− [Π][Π2] is 𝐹 ′(0) = 0, which removes the mass term at thesame time. If 𝐹 (0) = 0 but 𝐹 ′(0) = 0, the theory is massless about the specific reference metric,but infinitely strongly coupled about other backgrounds.

Instead to prevent the presence of the BD ghost fully non-linearly (or equivalently about anybackground), one should construct the mass term (or rather potential term) in such a way, thatall the higher derivative operators involving the helicity-0 mode (𝜕2𝜋)𝑛 are total derivatives. Thisis precisely what is achieved in the “ghost-free” model of massive gravity presented in Part II. Inthe next Part I we shall use higher dimensional GR to get some insight and intuition on how toconstruct a consistent theory of massive gravity.



26 Claudia de Rham

Part I

Massive Gravity from Extra Dimensions

3 Higher-Dimensional Scenarios

As seen in Section 2.5, the ‘most natural’ non-linear extension of the Fierz–Pauli mass term bears aghost. Constructing consistent theories of massive gravity has actually been a challenging task foryears, and higher-dimensional scenario can provide excellent frameworks for explicit realizations ofmassive gravity. The main motivation behind relying on higher dimensional gravity is twofold:

∙ The five-dimensional theory is explicitly covariant.

∙ A massless spin-2 field in five dimensions has five degrees of freedom which corresponds to thecorrect number of dofs for a massive spin-2 field in four dimensions without the pathologicalBD ghost.

While string theory and other higher dimensional theories give rise naturally to massive gravitons,they usually include a massless zero-mode. Furthermore, in the simplest models, as soon as thefirst massive mode is relevant so is an infinite tower of massive (Kaluza–Klein) modes and one isnever in a regime where a single massive graviton dominates, or at least this was the situationuntil the Dvali–Gabadadze–Porrati model (DGP) [208, 209, 207], provided the first explicit modelof (soft) massive gravity, based on a higher-dimensional braneworld model.

In the DGP model the graviton has a soft mass in the sense that its propagator does not havea simple pole at fixed value 𝑚, but rather admits a resonance. Considering the Kallen–Lehmannspectral representation [331, 374], the spectral density function 𝜌(𝜇2) in DGP is of the form

𝜌DGP(𝜇2) ∼ 𝑚0

𝜋𝜇

1

𝜇2 +𝑚20

, (3.1)

and so DGP corresponds to a theory of massive gravity with a resonance with width Δ𝑚 ∼ 𝑚0

about 𝑚 = 0.In a Kaluza–Klein decomposition of a flat extra dimension we have, on the other hand, an

infinite tower of massive modes with spectral density function

𝜌KK(𝜇2) ∼

∞∑𝑛=0

𝛿(𝜇2 − (𝑛𝑚0)2) . (3.2)

We shall see in the section on deconstruction (5) how one can truncate this infinite tower byperforming a discretization in real space rather than in momentum space a la Kaluza–Klein, so asto obtain a theory of a single massive graviton

𝜌MG(𝜇2) ∼ 𝛿(𝜇2 −𝑚2

0) , (3.3)

or a theory of multi-gravity (with 𝑁 -interacting gravitons),

𝜌multi−gravity(𝜇2) ∼

𝑁∑𝑛=0

𝛿(𝜇2 − (𝑛𝑚0)2) . (3.4)

In this language, bi-gravity is the special case of multi-gravity where 𝑁 = 2. These differentspectral representations, together with the cascading gravity extension of DGP are represented inFigure 1.



Massive Gravity 27

Figure 1: Spectral representation of different models. (a) DGP (b) higher-dimensional cascading gravityand (c) multi-gravity. Bi-gravity is the special case of multi-gravity with one massless mode and onemassive mode. Massive gravity is the special case where only one massive mode couples to the rest of thestandard model and the other modes decouple. (a) and (b) are models of soft massive gravity where thegraviton mass can be thought of as a resonance.

Recently, another higher dimensional embedding of bi-gravity was proposed in Ref. [495].Rather than performing a discretization of the extra dimension, the idea behind this model isto consider a two-brane DGP model, where the radion or separation between these branes is stabi-lized via a Goldberger-Wise stabilization mechanism [255] where the brane and the bulk include aspecific potential for the radion. At low energy the mass spectrum can be truncated to a masslessmode and a massive mode, reproducing a bi-gravity theory. However, the stabilization mechanisminvolves a relatively low scale and the correspondence breaks down above it. Nevertheless, thisprovides a first proof of principle for how to embed such a model in a higher-dimensional picturewithout discretization and could be useful to tackle some of the open questions of massive gravity.

In what follows we review how five-dimensional gravity is a useful starting point in order togenerate consistent four-dimensional theories of massive gravity, either for soft-massive gravity ala DGP and its extensions, or for hard massive gravity following a deconstruction framework.

The DGP model has played the role of a precursor for many developments in modified andmassive gravity and it is beyond the scope of this review to summarize all of them. In this reviewwe briefly summarize the DGP model and some key aspects of its phenomenology, and refer thereader to other reviews (see for instance [232, 390, 234]) for more details on the subject.

In this section, 𝐴,𝐵,𝐶 · · · = 0, . . . , 4 represent five-dimensional spacetime indices and 𝜇, 𝜈, 𝛼 · · · =0, . . . , 3 label four-dimensional spacetime indices. 𝑦 = 𝑥4 represents the fifth additional dimension,{𝑥𝐴} = {𝑥𝜇, 𝑦}. The five-dimensional metric is given by (5)𝑔𝐴𝐵(𝑥, 𝑦) while the four-dimensionalmetric is given by 𝑔𝜇𝜈(𝑥). The five-dimensional scalar curvature is (5)𝑅[𝐺] while 𝑅 = 𝑅[𝑔] is thefour-dimensional scalar-curvature. We use the same notation for the Einstein tensor where (5)𝐺𝐴𝐵is the five-dimensional one and 𝐺𝜇𝜈 represents the four-dimensional one built out of 𝑔𝜇𝜈 .

When working in the Einstein–Cartan formalism of gravity, a, b, c, · · · label five-dimensionalLorentz indices and 𝑎, 𝑏, 𝑐 · · · label the four-dimensional ones.

4 The Dvali–Gabadadze–Porrati Model

The idea behind the DGP model [209, 208, 207] is to start with a four-dimensional braneworldin an infinite size-extra dimension. A priori gravity would then be fully five-dimensional, withrespective Planck scale 𝑀5, but the matter fields localized on the brane could lead to an inducedcurvature term on the brane with respective Planck scale 𝑀Pl. See [22] for a potential embeddingof this model within string theory.

At small distances the induced curvature dominates and gravity behaves as in four dimensions,



28 Claudia de Rham

while at large distances the leakage of gravity within the extra dimension weakens the force ofgravity. The DGP model is thus a model of modified gravity in the infrared, and as we shall see,the graviton effectively acquires a soft mass, or resonance.

4.1 Gravity induced on a brane

We start with the five-dimensional action for the DGP model [209, 208, 207] with a brane localizedat 𝑦 = 0,

𝑆 =

∫d4𝑥 d𝑦

(𝑀3

5

4

√−(5)𝑔 (5)𝑅+ 𝛿(𝑦)

[√−𝑔𝑀

2Pl

2𝑅[𝑔] + ℒ𝑚(𝑔, 𝜓𝑖)

]), (4.1)

where 𝜓𝑖 represent matter field species confined to the brane with stress-energy tensor 𝑇𝜇𝜈 . Thisbrane is considered to be an orbifold brane enjoying a Z2-orbifold symmetry (so that the physicsat 𝑦 < 0 is the mirror copy of that at 𝑦 > 0.) We choose the convention where we consider−∞ < 𝑦 < ∞, reason why we have a factor or 𝑀3

5 /4 rather than 𝑀35 /2 if we had only consider

one side of the brane, for instance 𝑦 ≥ 0.The five-dimensional Einstein equation of motion are then given by

𝑀35

(5)𝐺𝐴𝐵 = 2𝛿(𝑦)(5)𝑇𝐴𝐵 (4.2)

with(5)𝑇𝐴𝐵 =

(−𝑀2

Pl𝐺𝜇𝜈 + 𝑇𝜇𝜈)𝛿𝜇𝐴𝛿

𝜈𝐵 . (4.3)

The Israel matching condition on the brane [323] can be obtained by integrating this equation over∫ 𝜀−𝜀 d𝑦 and taking the limit 𝜀→ 0, so that the jump in the extrinsic curvature across the brane isrelated to the Einstein tensor and stress-energy tensor of the matter field confined on the brane.

4.1.1 Perturbations about flat spacetime

In DGP the four-dimensional graviton is effectively massive. To see this explicitly, we look atperturbations about flat spacetime

d𝑠25 = (𝜂𝐴𝐵 + ℎ𝐴𝐵(𝑥, 𝑦)) d𝑥𝐴 d𝑥𝐵 . (4.4)

Since at this level we are dealing with five-dimensional GR, we are free to set the five-dimensionalgauge of our choice and choose five-dimensional de Donder gauge (a discussion about the brane-bending mode will follow)

𝜕𝐴ℎ𝐴𝐵 =

1

2𝜕𝐵ℎ

𝐴𝐴 . (4.5)

In this gauge the five-dimensional Einstein tensor is simply

(5)𝐺𝐴𝐵 = −1

225

(ℎ𝐴𝐵 −

1

2ℎ𝐶𝐶𝜂𝐴𝐵

), (4.6)

where 25 = 2+ 𝜕2𝑦 is the five-dimensional d’Alembertian and 2 is the four-dimensional one.

Since there is no source along the 𝜇𝑦 or 𝑦𝑦 directions ((5)𝑇𝜇𝑦 = 0 = (5)𝑇𝑦𝑦), we can immediatelyinfer that

25ℎ𝜇𝑦 = 0 ⇒ ℎ𝜇𝑦 = 0 (4.7)

25

(ℎ𝑦𝑦 − ℎ𝜇𝜇

)= 0 ⇒ ℎ𝑦𝑦 = ℎ𝜇𝜇 , (4.8)

up to an homogeneous mode which in this setup we set to zero. This does not properly account forthe brane-bending mode but for the sake of this analysis it will give the correct expression for the



Massive Gravity 29

metric fluctuation ℎ𝜇𝜈 . We will see in Section 4.2 how to keep track of the brane-bending modewhich is partly encoded in ℎ𝑦𝑦.

Using these relations in the five-dimensional de Donder gauge, we deduce the relation for thepurely four-dimensional part of the metric perturbation,

𝜕𝜇ℎ𝜇𝜈 = 𝜕𝜈ℎ

𝜇𝜇 . (4.9)

Using these relations in the projected Einstein equation, we get

1

2𝑀3

5

[2+ 𝜕2𝑦

](ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈) = −𝛿(𝑦)

(2𝑇𝜇𝜈 +𝑀2

Pl (2ℎ𝜇𝜈 − 𝜕𝜇𝜕𝜈ℎ)), (4.10)

where ℎ ≡ ℎ𝛼𝛼 = 𝜂𝜇𝜈ℎ𝜇𝜈 is the four-dimensional trace of the perturbations.Solving this equation with the requirement that ℎ𝜇𝜈 → 0 as 𝑦 → ±∞, we infer the following

profile for the perturbations along the extra dimension

ℎ𝜇𝜈(𝑥, 𝑦) = 𝑒−|𝑦|√−2ℎ𝜇𝜈(𝑥) , (4.11)

where the 2 should really be thought in Fourier space, and ℎ𝜇𝜈(𝑥) is set from the boundaryconditions on the brane. Integrating the Einstein equation across the brane, from −𝜀 to +𝜀, weget

1

2lim𝜀→0

𝑀35 [𝜕𝑦ℎ𝜇𝜈(𝑥, 𝑦)− ℎ(𝑥, 𝑦)𝜂𝜇𝜈 ]𝜀−𝜀 +𝑀2

Pl (2ℎ𝜇𝜈(𝑥, 0)− 𝜕𝜇𝜕𝜈ℎ(𝑥, 0))

= −2𝑇𝜇𝜈(𝑥) , (4.12)

yielding the modified linearized Einstein equation on the brane

𝑀2Pl

[(2ℎ𝜇𝜈 − 𝜕𝜇𝜕𝜈ℎ)−𝑚0

√−2 (ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈)

]= −2𝑇𝜇𝜈 , (4.13)

where all the metric perturbations are the ones localized at 𝑦 = 0 and the constant mass scale 𝑚0

is given by

𝑚0 =𝑀3

5

𝑀2Pl

. (4.14)

Interestingly, we see the special Fierz–Pauli combination ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈 appearing naturally from thefive-dimensional nature of the theory. At this level, this corresponds to a linearized theory ofmassive gravity with a scale-dependent effective mass 𝑚2(2) = 𝑚0

√−2, which can be thought in

Fourier space, 𝑚2(𝑘) = 𝑚0𝑘. We could now follow the same procedure as derived in Section 2.2.3and obtain the expression for the sourced metric fluctuation on the brane

ℎ𝜇𝜈 = − 2

𝑀2Pl

1

2−𝑚0

√−2

(𝑇𝜇𝜈 −

1

3𝑇𝜂𝜇𝜈 +

1

3𝑚√−2

𝜕𝜇𝜕𝜈𝑇

), (4.15)

where 𝑇 = 𝜂𝜇𝜈𝑇𝜇𝜈 is the trace of the four-dimensional stress-energy tensor localized on the brane.This yields the following gravitational exchange amplitude between two conserved sources 𝑇𝜇𝜈 and𝑇 ′𝜇𝜈 ,

𝒜DGP𝑇𝑇 ′ =

∫d4𝑥 ℎ𝜇𝜈𝑇

′𝜇𝜈 =

∫d4𝑥 𝑇 ′𝜇𝜈 𝑓massive

𝜇𝜈𝛼𝛽

2−𝑚0

√−2

𝑇𝛼𝛽 , (4.16)

where the polarization tensor 𝑓massive𝜇𝜈𝛼𝛽 is the same as that given for Fierz–Pauli in (2.57) in terms of

𝑚0. In particular the polarization tensor includes the standard factor of −1/3𝑇𝜂𝜇𝜈 as opposed to−1/2𝑇𝜂𝜇𝜈 as would be the case in GR. This is again the manifestation of the vDVZ discontinuitywhich is cured by the Vainshtein mechanism as for Fierz–Pauli massive gravity. See [165] for theexplicit realization of the Vainshtein mechanism in DGP which is where it was first shown to workexplicitly.



30 Claudia de Rham

4.1.2 Spectral representation

In Fourier space the propagator for the graviton in DGP is given by

��massive𝜇𝜈𝛼𝛽 (𝑘) = 𝑓massive

𝜇𝜈𝛼𝛽 (𝑘,𝑚0) 𝒢(𝑘) , (4.17)

with the massive polarization tensor 𝑓massive defined in (2.58) and

𝒢(𝑘) = 1

𝑘2 +𝑚0𝑘, (4.18)

which can be written in the Kallen–Lehmann spectral representation as a sum of free propagatorswith mass 𝜇,

𝒢(𝑘) =∫ ∞

0

𝜌(𝜇2)

𝑘2 + 𝜇2d𝜇2 , (4.19)

with the spectral density 𝜌(𝜇2)

𝜌(𝜇2) =1

𝜋

𝑚0

𝜇

1

𝜇2 +𝑚20

, (4.20)

which is represented in Figure 1. As already emphasized, the graviton in DGP cannot be thoughtof a single massive mode, but rather as a resonance picked about 𝜇 = 0.

We see that the spectral density is positive for any 𝜇2 > 0, confirming the fact that about thenormal (flat) branch of DGP there is no ghost.

Notice as well that in the massless limit 𝑚0 → 0, we see appearing a representation of the Diracdelta function,

lim𝑚→0

1

𝜋

𝑚0

𝜇

1

𝜇2 +𝑚20

= 𝛿(𝜇2) , (4.21)

and so the massless mode is singled out in the massless limit of DGP (with the different tensor

structure given by 𝑓massive𝜇𝜈𝛼𝛽 = 𝑓

(0)𝜇𝜈𝛼𝛽 which is the origin of the vDVZ discontinuity see Section 2.2.3.)

4.2 Brane-bending mode

Five-dimensional gauge-fixing

In Section 4.1.1 we have remained vague about the gauge-fixing and the implications for the braneposition. The brane-bending mode is actually important to keep track of in DGP and we shall dothat properly in what follows by keeping all the modes.

We work in the five-dimensional ADM split with the lapse 𝑁 = 1/√𝑔𝑦𝑦 = 1 + 1

2ℎ𝑦𝑦, theshift 𝑁𝜇 = 𝑔𝜇𝑦 and the four-dimensional part of the metric, 𝑔𝜇𝜈(𝑥, 𝑦) = 𝜂𝜇𝜈 + ℎ𝜇𝜈(𝑥, 𝑦). Thefive-dimensional Einstein–Hilbert term is then expressed as

ℒ(5)R =

𝑀35

4

√−𝑔𝑁

(𝑅[𝑔] + [𝐾]2 − [𝐾2]

), (4.22)

where square brackets correspond to the trace of a tensor with respect to the four-dimensionalmetric 𝑔𝜇𝜈 and 𝐾𝜇𝜈 is the extrinsic curvature

𝐾𝜇𝜈 =1

2𝑁(𝜕𝑦𝑔𝜇𝜈 −𝐷𝜇𝑁𝜈 −𝐷𝜈𝑁𝜇) , (4.23)

and 𝐷𝜇 is the covariant derivative with respect to 𝑔𝜇𝜈 .



Massive Gravity 31

First notice that the five-dimensional de Donder gauge choice (4.5) can be made using thefive-dimensional gauge fixing term

ℒ(5)Gauge−Fixing = −𝑀

35

8

(𝜕𝐴ℎ

𝐴𝐵 −

1

2𝜕𝐵ℎ

𝐴𝐴

)2

(4.24)

= −𝑀35

8

[(𝜕𝜇ℎ

𝜇𝜈 −

1

2𝜕𝜈ℎ+ 𝜕𝑦𝑁𝜈 −

1

2𝜕𝜈ℎ𝑦𝑦

)2

(4.25)

+

(𝜕𝜇𝑁

𝜇 +1

2𝜕𝑦ℎ𝑦𝑦 −

1

2𝜕𝑦ℎ

)2],

where we keep the same notation as previously, ℎ = 𝜂𝜇𝜈ℎ𝜇𝜈 is the four-dimensional trace.After fixing the de Donder gauge (4.5), we can make the addition gauge transformation 𝑥𝐴 →

𝑥𝐴 + 𝜉𝐴, and remain in de Donder gauge provided 𝜉𝐴 satisfies linearly 25𝜉𝐴 = 0. This residual

gauge freedom can be used to further fix the gauge on the brane (see [389] for more details, weonly summarize their derivation here).

Four-dimensional Gauge-fixing

Keeping the brane at the fixed position 𝑦 = 0 imposes 𝜉𝑦 = 0 since we need 𝜉𝑦(𝑦 = 0) = 0 and 𝜉should be bounded as 𝑦 →∞ (the situation is slightly different in the self-accelerating branch andthis mode can lead to a ghost, see Section 4.4 as well as [361, 98]).

Using the bulk profile ℎ𝐴𝐵(𝑥, 𝑦) = 𝑒−√−2|𝑦|ℎ𝐴𝐵(𝑥) and integrating over the extra dimension,

we obtain the contribution from the bulk on the brane (including the contribution from the gauge-fixing term) in terms of the gauge invariant quantity

ℎ𝜇𝜈 = ℎ𝜇𝜈 +2√−2

𝜕(𝜇𝑁𝜈) = −2√−2

𝐾𝜇𝜈 (4.26)

𝑆integratedbulk =

𝑀35

4

∫d4𝑥

[− 1

2ℎ𝜇𝜈√−2

(ℎ𝜇𝜈 −

1

2ℎ𝜂𝜇𝜈

)+

1

2ℎ𝑦𝑦√−2

(ℎ− 1

2ℎ𝑦𝑦

)].

Notice again a factor of 2 difference from [389] which arises from the fact that we integrate from𝑦 = −∞ to 𝑦 = +∞ imposing a Z2-mirror symmetry at 𝑦 = 0, rather than considering only oneside of the brane as in [389]. Both conventions are perfectly reasonable.

The integrated bulk action (4.27) is invariant under the residual linearized gauge symmetry

ℎ𝜇𝜈 → ℎ𝜇𝜈 + 2𝜕(𝜇𝜉𝜈) (4.27)

𝑁𝜇 → 𝑁𝜇 −√−2𝜉𝜈 (4.28)

ℎ𝑦𝑦 → ℎ𝑦𝑦 (4.29)

which keeps both ℎ𝜇𝜈 and ℎ𝑦𝑦 invariant. The residual gauge symmetry can be used to set thegauge on the brane, and at this level from (4.27) we can see that the most convenient gauge fixingterm is [389]

ℒ(4)Residual Gauge−Fixing = −𝑀

2Pl

4

(𝜕𝜇ℎ

𝜇𝜈 −

1

2𝜕𝜈ℎ+𝑚0𝑁𝜈

)2

, (4.30)

with again𝑚0 =𝑀35 /𝑀

2Pl, so that the induced Lagrangian on the brane (including the contribution

from the residual gauge fixing term) is

𝑆boundary =𝑀2

Pl

4

∫d4𝑥

[1

2ℎ𝜇𝜈2(ℎ𝜇𝜈 −

1

2ℎ𝜂𝜇𝜈)− 2𝑚0𝑁𝜇

(𝜕𝛼ℎ

𝛼𝜇 − 1

2𝜕𝜇ℎ

)−𝑚2

0𝑁𝜇𝑁𝜇

]. (4.31)



32 Claudia de Rham

Combining the five-dimensional action from the bulk (4.27) with that on the boundary (4.31) weend up with the linearized action on the four-dimensional DGP brane [389]

𝑆(lin)DGP =

𝑀2Pl

4

∫d4𝑥

[1

2ℎ𝜇𝜈

[2−𝑚0

√−2](ℎ𝜇𝜈 −

1

2ℎ𝜂𝜇𝜈)−𝑚0𝑁

𝜇𝜕𝜇ℎ𝑦𝑦 (4.32)

−𝑚0𝑁𝜇[√−2+𝑚0

]𝑁𝜇 −

𝑚0

4ℎ𝑦𝑦√−2(ℎ𝑦𝑦 − 2ℎ)

].

As shown earlier we recover the theory of a massive graviton in four dimensions, with a soft mass𝑚2(2) = 𝑚0

√−2. This analysis has allowed us to keep track of the physical origin of all the modes

including the brane-bending mode which is especially relevant when deriving the decoupling limitas we shall see below.

The kinetic mixing between these different modes can be diagonalized by performing the changeof variables [389]

ℎ𝜇𝜈 =1

𝑀Pl

(ℎ′𝜇𝜈 + 𝜋𝜂𝜇𝜈

)(4.33)

𝑁𝜇 =1

𝑀Pl√𝑚0

𝑁 ′𝜇 +

1

𝑀Pl𝑚0𝜕𝜇𝜋 (4.34)

ℎ𝑦𝑦 = − 2√−2

𝑚0𝑀Pl𝜋 , (4.35)

so we see that the mode 𝜋 is directly related to ℎ𝑦𝑦. In the case of Section 4.1.1, we had set ℎ𝑦𝑦 = 0and the field 𝜋 is then related to the brane bending mode. In either case we see that the extrinsiccurvature 𝐾𝜇𝜈 carries part of this mode.

Omitting the mass terms and other relevant operators, the action is diagonalized in terms ofthe different graviton modes at the linearized level ℎ′𝜇𝜈 (which encodes the helicity-2 mode), 𝑁 ′

𝜇

(which is part of the helicity-1 mode) and 𝜋 (helicity-0 mode),

𝑆(lin)DGP =

1

4

∫d4𝑥

[1

2ℎ′𝜇𝜈2(ℎ′𝜇𝜈 −

1

2ℎ′𝜂𝜇𝜈)−𝑁 ′𝜇√−2𝑁 ′

𝜇 + 3𝜋2𝜋

]. (4.36)

Decoupling limit

We will be discussing the meaning of ‘decoupling limits’ in more depth in the context of multi-gravity and ghost-free massive gravity in Section 8. The main idea behind the decoupling limit isto separate the physics of the different modes. Here we are interested in following the interactionsof the helicity-0 mode without the complications from the standard helicity-2 interactions thatalready arise in GR. For this purpose we can take the limit 𝑀Pl → ∞ while simultaneouslysending 𝑚0 = 𝑀3

5 /𝑀2Pl → 0 while keeping the scale Λ = (𝑚2

0𝑀Pl)1/3 fixed. This is the scale at

which the first interactions arise in DGP.In DGP the decoupling limit should be taken by considering the full five-dimensional theory,

as was performed in [389]. The four-dimensional Einstein–Hilbert term does not give to anyoperators before the Planck scale, so in order to look for the irrelevant operator that come atthe lowest possible scale, it is sufficient to focus on the boundary term from the five-dimensionalaction. It includes operators of the form

ℒ(5)boundary ⊃ 𝑚0𝑀

2Pl𝜕

(ℎ′𝜇𝜈𝑀Pl

)𝑛(𝑁 ′𝜇√

𝑚0𝑀Pl

)𝑘 (𝜕𝜋

𝑚0𝑀Pl

)ℓ, (4.37)



Massive Gravity 33

with integer powers 𝑛, 𝑘, ℓ ≥ 0 and 𝑛+ 𝑘+ ℓ ≥ 3 since we are dealing with interactions. The scaleat which such an operator arises is

Λ𝑛,𝑘,ℓ =(𝑀𝑛+𝑘+ℓ−2

Pl 𝑚𝑘/2+ℓ−10

)1/(𝑛+3𝑘/2+2ℓ−3)

(4.38)

and it is easy to see that the lowest possible scale is Λ3 = (𝑀Pl𝑚20)

1/3 which arises for 𝑛 = 0, 𝑘 = 0and ℓ = 3, it is thus a cubic interaction in the helicity-0 mode 𝜋 which involves four derivatives.Since it is only a cubic interaction, we can scan all the possible ways 𝜋 enters at the cubic levelin the five-dimensional Einstein–Hilbert action. The relevant piece are the ones from the extrinsiccurvature in (4.22), and in particular the combination 𝑁([𝐾]2 − [𝐾2]), with

𝑁 = 1 +1

2𝑒−

√−2𝑦ℎ𝑦𝑦 (4.39)

𝐾𝜇𝜈 = −1

2(1− 1

2𝑒−

√−2𝑦ℎ𝑦𝑦)(𝜕𝜇𝑁𝜈 + 𝜕𝜈𝑁𝜇) . (4.40)

Integrating 𝑚0𝑀2Pl𝑁([𝐾]2 − [𝐾2]) along the extra dimension, we obtain the cubic contribution in

𝜋 on the brane (using the relations (4.34) and (4.35))

ℒΛ3=

1

2Λ33

(𝜕𝜋)22𝜋 . (4.41)

So the decoupling limit of DGP arises at the scale Λ3 and reduces to a cubic Galileon for thehelicity-0 mode with no interactions for the helicity-2 and -1 modes,

ℒDL DGP =1

8ℎ′𝜇𝜈2

(ℎ′𝜇𝜈 −

1

2ℎ′𝜂𝜇𝜈

)− 1

4𝑁 ′𝜇√−2𝑁 ′

𝜇 (4.42)

+3

2𝜋2𝜋 +

1

2Λ33

(𝜕𝜋)22𝜋 .

4.3 Phenomenology of DGP

The phenomenology of DGP is extremely rich and has led to many developments. In what followswe review one of the most important implications of the DGP for cosmology which the existenceof self-accelerating solutions. The cosmology and phenomenology of DGP was first derived in [159,163] (see also [388, 385, 387, 386]).

4.3.1 Friedmann equation in de Sitter

To get some intuition on how cosmology gets modified in DGP, we first look at de Sitter-likesolutions and then infer the full Friedmann equation in a FLRW-geometry. We thus start withfive-dimensional Minkowski in de Sitter slicing (this can be easily generalized to FLRW-slicing),

d𝑠25 = 𝑏2(𝑦)(d𝑦2 + 𝛾(dS)𝜇𝜈 d𝑥𝜇 d𝑥𝜈

), (4.43)

where 𝛾(dS)𝜇𝜈 is the four-dimensional de Sitter metric with constant Hubble parameter 𝐻,

𝛾(dS)𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = − d𝑡2 + 𝑎2(𝑡) d𝑥2, and the scale factor is given by 𝑎(𝑡) = exp(𝐻𝑡). The met-

ric (4.43) is indeed Minkowski in de Sitter slicing if the warp factor 𝑏(𝑦) is given by

𝑏(𝑦) = 𝑒𝜖𝐻|𝑦| , with 𝜖 = ±1 , (4.44)



34 Claudia de Rham

and the mod 𝑦 has be imposed by the Z2-orbifold symmetry. As we shall see the branch 𝜖 = +1corresponds to the self-accelerating branch of DGP and 𝜖 = −1 is the stable, normal branch ofDGP.

We can now derive the Friedmann equation on the brane by integrating over the 00-componentof the Einstein equation (4.2) with the source (4.3) and consider some energy density 𝑇00 = 𝜌. Thefour-dimensional Einstein tensor gives the standard contribution 𝐺00 = 3𝐻2 on the brane and sowe obtain the modified Friedmann equation

𝑀35

2

[lim𝜀→0

∫ 𝜀

−𝜀

(5)𝐺00 d𝑦

]+ 3𝑀2

Pl𝐻2 = 𝜌 , (4.45)

with (5)𝐺00 = 3(𝐻2 − 𝑏′′(𝑦)/𝑏(𝑦)), so

lim𝜀→0

∫ 𝜀

−𝜀

(5)𝐺00 d𝑦 = −6𝜖𝐻 , (4.46)

leading to the modified Friedmann equation,

𝐻2 − 𝜖𝑚0𝐻 =1

3𝑀2Pl

𝜌 , (4.47)

where the five-dimensional nature of the theory is encoded in the new term −𝜖𝑚0𝐻 (this newcontribution can be seen to arise from the helicity-0 mode of the graviton and could have beenderived using the decoupling limit of DGP.)

For reasons which will become clear in what follows, the choice 𝜖 = −1 corresponds to the stablebranch of DGP while the other choice 𝜖 = +1 corresponds to the self-accelerating branch of DGP.As is already clear from the higher-dimensional perspective, when 𝜖 = +1, the warp factor growsin the bulk (unless we think of the junction conditions the other way around), which is alreadysignaling towards a pathology for that branch of solution.

4.3.2 General Friedmann equation

This modified Friedmann equation has been derived assuming a constant 𝐻, which is only con-sistent if the energy density is constant (i.e., a cosmological constant). We can now derive thegeneralization of this Friedmann equation for non-constant 𝐻. This amounts to account for �� andother derivative corrections which might have been omitted in deriving this equation by assumingthat 𝐻 was constant. But the Friedmann equation corresponds to the Hamiltonian constraintequation and higher derivatives (e.g., �� ⊃ �� and higher derivatives of 𝐻) would imply that thisequation is no longer a constraint and this loss of constraint would imply that the theory admitsa new degree of freedom about generic backgrounds namely the BD ghost (see the discussion ofSection 7).

However, in DGP we know that the BD ghost is absent (this is ensured by the five-dimensionalnature of the theory, in five dimensions we start with five dofs, and there is thus no sixth BDmode). So the Friedmann equation cannot include any derivatives of 𝐻, and the Friedmannequation obtained assuming a constant 𝐻 is actually exact in FLRW even if 𝐻 is not constant.So the constraint (4.47) is the exact Friedmann equation in DGP for any energy density 𝜌 on thebrane.

The same trick can be used for massive gravity and bi-gravity and the Friedmann equa-tions (12.51), (12.52) and (12.54) are indeed free of any derivatives of the Hubble parameter.



Massive Gravity 35

4.3.3 Observational viability of DGP

Independently of the ghost issue in the self-accelerating branch of the model, there has been a vastamount of investigation on the observational viability of both the self-accelerating branch and thenormal (stable) branch of DGP. First because many of these observations can apply equally wellto the stable branch of DGP (modulo a minus sign in some of the cases), and second and foremostbecause DGP represents an excellent archetype in which ideas of modified gravity can be tested.

Observational tests of DGP fall into the following two main categories:

∙ Tests of the Friedmann equation. This test was performed mainly using Supernovae, butalso using Baryonic Acoustic Oscillations and the CMB so as to fix the background historyof the Universe [162, 217, 221, 286, 391, 23, 405, 481, 304, 382, 462]. Current observationsseem to slightly disfavor the additional 𝐻 term in the Friedmann equation of DGP, even inthe normal branch where the late-time acceleration of the Universe is due to a cosmologicalconstant as in ΛCDM. These put bounds on the graviton mass in DGP to the order of𝑚0 . 10−1𝐻0, where 𝐻0 is the Hubble parameter today (see Ref. [492] for the latest boundsat the time of writing, including data from Planck). Effectively this means that in orderfor DGP to be consistent with observations, the graviton mass can have no effect on thelate-time acceleration of the Universe.

∙ Tests of an extra fifth force, either within the solar system, or during structure formation(see for instance [362, 260, 452, 451, 222, 482] Refs. [453, 337, 442] for N-body simulationsas well as Ref. [17, 441] using weak lensing).

Evading fifth force experiments will be discussed in more detail within the context of theVainshtein mechanism in Section 10.1 and thereafter, and we save the discussion to thatsection. See Refs. [388, 385, 387, 386, 444] for a five-dimensional study dedicated to DGP.The study of cosmological perturbations within the context of DGP was also performed indepth for instance in [367, 92].

4.4 Self-acceleration branch

The cosmology of DGP has led to a major conceptual breakthrough, namely the realization thatthe Universe could be ‘self-accelerating’. This occurs when choosing the 𝜖 = +1 branch of DGP,the Friedmann equation in the vacuum reduces to [159, 163]

𝐻2 −𝑚0𝐻 = 0 , (4.48)

which admits a non-trivial solution 𝐻 = 𝑚0 in the absence of any cosmological constant norvacuum energy. In itself this would not solve the old cosmological constant problem as the vacuumenergy ought to be set to zero on its own, but it can lead to a model of ‘dark gravity’ where theamount of acceleration is governed by the scale 𝑚0 which is stable against quantum corrections.

This realization has opened a new field of study in its own right. It is beyond the scope ofthis review on massive gravity to summarize all the interesting developments that arose in thepast decade and we simply focus on a few elements namely the presence of a ghost in this self-accelerating branch as well as a few cosmological observations.

Ghost

The existence of a ghost on the self-accelerating branch of DGP was first pointed out in thedecoupling limit [389, 411], where the helicity-0 mode of the graviton is shown to enter with thewrong sign kinetic in this branch of solutions. We emphasize that the issue of the ghost in theself-accelerating branch of DGP is completely unrelated to the sixth BD ghost on some theories



36 Claudia de Rham

of massive gravity. In DGP there are five dofs one of which is a ghost. The analysis was thengeneralized in the fully fledged five-dimensional theory by K. Koyama in [360] (see also [263, 361]and [98]).

When perturbing about Minkowski, it was shown that the graviton has an effective mass𝑚2 = 𝑚0

√−2. When perturbing on top of the self-accelerating solution a similar analysis can be

performed and one can show that in the vacuum the graviton has an effective mass at precisely theHiguchi-bound, 𝑚2

eff = 2𝐻2 (see Ref. [307]). When matter or a cosmological constant is includedon the brane, the graviton mass shifts either inside the forbidden Higuchi-region 0 < 𝑚2

eff < 2𝐻2,or outside 𝑚2

eff > 2𝐻2. We summarize the three case scenario following [360, 98]

∙ In [307] it was shown that when the effective mass is within the forbidden Higuchi-region,the helicity-0 mode of graviton has the wrong sign kinetic term and is a ghost.

∙ Outside this forbidden region, when𝑚2eff > 2𝐻2, the zero-mode of the graviton is healthy but

there exists a new normalizable brane-bending mode in the self-accelerating branch8 whichis a genuine degree of freedom. For 𝑚2

eff > 2𝐻2 the brane-bending mode was shown to be aghost.

∙ Finally, at the critical mass 𝑚2eff = 2𝐻2 (which happens when no matter nor cosmological

constant is present on the brane), the brane-bending mode takes the role of the helicity-0mode of the graviton, so that the theory graviton still has five degrees of freedom, and thismode was shown to be a ghost as well.

In summary, independently of the matter content of the brane, so long as the graviton is massive𝑚2

eff > 0, the self-accelerating branch of DGP exhibits a ghost. See also [210] for an exact non-perturbative argument studying domain walls in DGP. In the self-accelerating branch of DGPdomain walls bear a negative gravitational mass. This non-perturbative solution can also be usedas an argument for the instability of that branch.

Evading the ghost?

Different ways to remove the ghosts were discussed for instance in [325] where a second brane wasincluded. In this scenario it was then shown that the graviton could be made stable but at thecost of including a new spin-0 mode (that appears as the mode describing the distance betweenthe branes).

Alternatively it was pointed out in [233] that if the sign of the extrinsic curvature was flipped,the self-accelerating solution on the brane would be stable.

Finally, a stable self-acceleration was also shown to occur in the massless case𝑚2eff = 0 by relying

on Gauss–Bonnet terms in the bulk and a self-source AdS5 solution [156]. The five-dimensionaltheory is then similar as that of DGP (4.1) but with the addition of a five-dimensional Gauss–Bonnet term ℛ2

GB in the bulk and the wrong sign five-dimensional Einstein–Hilbert term,

𝑆 =

∫d5𝑥

[√−(5)𝑔

(−𝑀

35

4(5)𝑅[(5)𝑔]− 𝑀3

5 ℓ2

4(5)ℛ2

GB[(5)𝑔]

)(4.49)

+𝛿(𝑦)

[√−𝑔𝑀

2Pl

2𝑅+ ℒ𝑚(𝑔, 𝜓𝑖)

]].

The idea is not so dissimilar as in new massive gravity (see Section 13), where here the wrongsign kinetic term in five-dimensions is balanced by the Gauss–Bonnet term in such a way that the

8 In the normal branch of DGP, this brane-bending mode turns out not to be normalizable. The normalizablebrane-bending mode which is instead present in the normal branch fully decouples and plays no role.



Massive Gravity 37

graviton has the correct sign kinetic term on the self-sourced AdS5 solution. The length scale ℓis related to this AdS length scale, and the self-accelerating branch admits a stable (ghost-free)de Sitter solution with 𝐻 ∼ ℓ−1.

We do not discuss this model any further in what follows since the graviton admits a zero(massless) mode. It is feasible that this model can be understood as a bi-gravity theory where themassive mode is a resonance. It would also be interesting to see how this model fits in with theGalileon theories [412] which admit stable self-accelerating solution.

In what follows, we go back to the standard DGP model be it the self-accelerating branch(𝜖 = 1) or the normal branch (𝜖 = −1).

4.5 Degravitation

One of the main motivations behind modifying gravity in the infrared is to tackle the old cosmolog-ical constant problem. The idea behind ‘degravitation’ [211, 212, 26, 216] is if gravity is modifiedin the IR, then a cosmological constant (or the vacuum energy) could have a smaller impact on thegeometry. In these models, we would live with a large vacuum energy (be it at the TeV scale or atthe Planck scale) but only observe a small amount of late-acceleration due to the modification ofgravity. In order for a theory of modified gravity to potentially tackle the old cosmological constantproblem via degravitation it needs to have the two following properties:

1. First, gravity must be weaker in the infrared and effectively massive [216] so that the effectof IR sources can be degravitated.

2. Second, there must exist some (nearly) static attractor solutions towards which the systemcan evolve at late-time for arbitrary value of the vacuum energy or cosmological constant.

Flat solution with a cosmological constant

The first requirement is present in DGP, but as was shown in [216] in DGP gravity is not ‘sufficientlyweak’ in the IR to allow degravitation solutions. Nevertheless, it was shown in [164] that the normalbranch of DGP satisfies the second requirement for any negative value of the cosmological constant.In these solutions the five-dimensional spacetime is not Lorentz invariant, but in a way which wouldnot (at this background level) be observed when confined on the four-dimensional brane.

For positive values of the cosmological constant, DGP does not admit a (nearly) static solution.This can be understood at the level of the decoupling limit using the arguments of [216] andgeneralized for other mass operators.

Inspired by the form of the graviton in DGP, 𝑚2(2) = 𝑚0

√−2, we can generalize the form of

the graviton mass to

𝑚2(2) = 𝑚20

(−2𝑚2

0

)𝛼, (4.50)

with 𝛼 a positive dimensionless constant. 𝛼 = 1 corresponds to a modification of the kinetic term.As shown in [153], any such modification leads to ghosts, so we do not consider this case here.𝛼 > 1 corresponds to a UV modification of gravity, and so we focus on 𝛼 < 1.

In the decoupling limit the helicity-2 decouples from the helicity-0 mode which behaves (sym-bolically) as follows [216]

32𝜋 − 1

𝑀Pl𝑚4(1−𝛼)0

2(21−𝛼𝜋

)2+ · · · = − 1

𝑀Pl𝑇 , (4.51)

where 𝑇 is the trace of the stress-energy tensor of external matter fields. At the linearized level,matter couples to the metric 𝑔𝜇𝜈 = 𝜂𝜇𝜈+

1𝑀Pl

(ℎ′𝜇𝜈+𝜋𝜂𝜇𝜈). We now check under which conditions we



38 Claudia de Rham

can still recover a nearly static metric in the presence of a cosmological constant 𝑇𝜇𝜈 = −ΛCC𝑔𝜇𝜈 .In the linearized limit of GR this leads to the profile for the helicity-2 mode (which in that casecorresponds to a linearized de Sitter solution)

ℎ′𝜇𝜈 = − ΛCC

6𝑀Pl𝜂𝜌𝜎𝑥

𝜌𝑥𝜎𝜂𝜇𝜈 . (4.52)

One way we can obtain a static solution in this extended theory of massive gravity at the linearlevel is by ensuring that the solution for 𝜋 cancels out that of ℎ′𝜇𝜈 so that the metric 𝑔𝜇𝜈 remains

flat. 𝜋 = + ΛCC

6𝑀Pl𝜂𝜇𝜈𝑥

𝜇𝑥𝜈 is actually the solution of (4.51) when only the term 32𝜋 contributes

and all the other operators vanish for 𝜋 ∝ 𝑥𝜇𝑥𝜇. This is the case if 𝛼 < 1/2 as shown in [216].

This explains why in the case of DGP which corresponds to border line scenario 𝛼 = 1/2, one cannever fully degravitate a cosmological constant.

Extensions

This realization has motivated the search for theories of massive gravity with 0 ≤ 𝛼 < 1/2, andespecially the extension of DGP to higher dimensions where the parameter 𝛼 can get as closeto zero as required. This is the main motivation behind higher dimensional DGP [359, 240] andcascading gravity [135, 148, 132, 149] as we review in what follows. (In [433] it was also shown howa regularized version of higher dimensional DGP could be free of the strong coupling and ghostissues).

Note that 𝛼 ≡ 0 corresponds to a hard mass gravity. Within the context of DGP, such a modelwith an ‘auxiliary’ extra dimension was proposed in [235, 133] where we consider a finite-size largeextra dimension which breaks five-dimensional Lorentz invariance. The five-dimensional actionis motivated by the five-dimensional gravity with scalar curvature in the ADM decomposition(5)𝑅 = 𝑅[𝑔] + [𝐾]2 − [𝐾2], but discarding the contribution from the four-dimensional curvature𝑅[𝑔]. Similarly as in DGP, the four-dimensional curvature still appears induced on the brane

𝑆 =𝑀2

Pl

2

∫ ℓ

0

d𝑦

∫d4𝑥√−𝑔(𝑚0

([𝐾]2 − [𝐾2]

)+ 𝛿(𝑦)𝑅[𝑔]

), (4.53)

where ℓ is the size of the auxiliary extra dimension and 𝑔𝜇𝜈 is a four-dimensional metric and we setthe lapse to one (this shift can be kept and will contribute to the four-dimensional Stuckelberg fieldwhich restores four-dimensional invariance, but at this level it is easier to work in the gauge wherethe shift is set to zero and reintroduce the Stuckelberg fields directly in four dimensions). Imposingthe Dirichlet conditions 𝑔𝜇𝜈(𝑥, 𝑦 = 0) = 𝑓𝜇𝜈 , we are left with a theory of massive gravity at 𝑦 = 0,with reference metric 𝑓𝜇𝜈 and hard mass 𝑚0. Here again the special structure

([𝐾]2 − [𝐾2]

)inherited (or rather inspired) from five-dimensional gravity ensures the Fierz–Pauli structure andthe absence of ghost at the linearized level. Up to cubic order in perturbations it was shown in [138]that the theory is free of ghost and its decoupling limit is that of a Galileon.

Furthermore, it was shown in [133] that it satisfies both requirements presented above to po-tentially help degravitating a cosmological constant. Unfortunately at higher orders this modelis plagued with the BD ghost [291] unless the boundary conditions are chosen appropriately [59].For this reason we will not review this model any further in what follows and focus instead on theghost-free theory of massive gravity derived in [137, 144]

4.5.1 Cascading gravity

Deficit angle

It is well known that a tension on a cosmic string does not cause the cosmic strong to inflate butrather creates a deficit angle in the two spatial dimensions orthogonal to the string. Similarly, if



Massive Gravity 39

we consider a four-dimensional brane embedded in six-dimensional gravity, then a tension on thebrane leads to the following flat geometry

d𝑠26 = 𝜂𝜇𝜈 d𝑥𝜇 d𝑥𝜈 + d𝑟2 + 𝑟2(1− Δ𝜃

2𝜋

)d𝜃2 , (4.54)

where the two extra dimensions are expressed in polar coordinates {𝑟, 𝜃} and Δ𝜃 is a constantwhich parameterize the deficit angle in this canonical geometry. This deficit angle is related to thetension on the brane ΛCC and the six-dimensional Planck scale (assuming six-dimensional gravity)

Δ𝜃 = 2𝜋ΛCC

𝑀46

. (4.55)

For a positive tension ΛCC > 0, this creates a positive deficit angle and since Δ𝜃 cannot be largerthan 2𝜋, the maximal tension on the brane is𝑀4

6 . For a negative tension, on the other hand, thereis no such bound as it creates a surplus of angle, see Figure 2.

Figure 2: Codimension-2 brane with positive (resp. negative) tension brane leading to a positive (resp.negative) deficit angle in the two extra dimensions.

This interesting feature has lead to many potential ways to tackle the cosmological constantby considering our Universe to live in a 3 + 1-dimensional brane embedded in two or more largeextra dimensions. (See Refs. [4, 3, 408, 414, 80, 470, 458, 459, 86, 82, 247, 333, 471, 81, 426, 409,373, 85, 460, 155] for the supersymmetric large-extra-dimension scenario as an alternative way to



40 Claudia de Rham

tackle the cosmological constant problem). Extending the DGP to more than one extra dimensioncould thus provide a natural way to tackle the cosmological constant problem.

Spectral representation

Furthermore in 𝑛-extra dimensions the gravitational potential is diluted as 𝑉 (𝑟) ∼ 𝑟−1−𝑛. Ifthe propagator has a Kallen–Lehmann spectral representation with spectral density 𝜌(𝜇2), theNewtonian potential has the following spectral representation

𝑉 (𝑟) =

∫ ∞

0

𝜌(𝜇2)𝑒−𝜇𝑟

𝑟d𝜇2 . (4.56)

In a higher-dimensional DGP scenario, the gravitation potential behaves higher dimensional atlarge distance, 𝑉 (𝑟) ∼ 𝑟−(1+𝑛) which implies 𝜌(𝜇2) ∼ 𝜇𝑛−2 in the IR as depicted in Figure 1.

Working back in terms of the spectral representation of the propagator as given in (4.19),this means that the propagator goes to 1/𝑘 in the IR as 𝜇 → 0 when 𝑛 = 1 (as we know fromDGP), while it goes to a constant for 𝑛 > 1. So for more than one extra dimension, the theorytends towards that of a hard mass graviton in the far IR, which corresponds to 𝛼 → 0 in theparametrization of (4.50). Following the arguments of [216] such a theory should thus be a goodcandidate to tackle the cosmological constant problem.

A brane on a brane

Both the spectral representation and the fact that codimension-two (and higher) branes can accom-modate for a cosmological constant while remaining flat has made the field of higher-codimensionbranes particularly interesting.

However, as shown in [240] and [135, 148, 132, 149], the straightforward extension of DGP totwo large extra dimensions leads to ghost issues (sixth mode with the wrong sign kinetic term, seealso [290, 70]) as well as divergences problems (see Refs. [256, 131, 130, 423, 422, 355, 83]).

To avoid these issues, one can consider simply applying the DGP procedure step by step andconsider a 4+ 1-dimensional DGP brane embedded in six dimension. Our Universe would then beon a 3 + 1-dimensional DGP brane embedded in the 4 + 1 one, (note we only consider one side ofthe brane here which explains the factor of 2 difference compared with (4.1))

𝑆 =𝑀4

6

2

∫d6𝑥√−𝑔6 (6)𝑅+

𝑀35

2

∫d5𝑥√−𝑔5 (5)𝑅 (4.57)

+𝑀2

Pl

2

∫d4𝑥√−𝑔4 (4)𝑅+

∫d4𝑥ℒmatter(𝑔4, 𝜓) .

This model has two cross-over scales: 𝑚5 = 𝑀35 /𝑀

2Pl which characterizes the scale at which one

crosses from the four-dimensional to the five-dimensional regime, and 𝑚6 = 𝑀46 /𝑀

35 yielding the

crossing from a five-dimensional to a six-dimensional behavior. Of course we could also have asimultaneous crossing if 𝑚5 = 𝑚6. In what follows we focus on the case where 𝑀Pl > 𝑀5 > 𝑀6.

Performing the same linearized analysis as in Section 4.1.1 we can see that the four-dimensionaltheory of gravity is effectively massive with the soft mass in Fourier space

𝑚2(𝑘) =𝜋𝑚5

4

√𝑚2

6 − 𝑘2

arcth√

𝑚6−𝑘𝑚6+𝑘

. (4.58)

We see that the 4 + 1-dimensional brane plays the role of a regulator (a divergence occurs in thelimit 𝑚5 → 0).



Massive Gravity 41

In this six-dimensional model, there are effectively two new scalar degrees of freedom (arisingfrom the extra dimensions). We can ensure that both of them have the correct sign kinetic termby

∙ Either smoothing out the brane [240, 148] (this means that one should really consider asix-dimensional curvature on both the smoothed 4 + 1 and on the 3 + 1-dimensional branes,which is something one would naturally expect9).∙ Or by including some tension on the 3 + 1 brane (which is also something natural sincethe setup is designed to degravitate a large cosmological constant on that brane). This wasshown to be ghost free in the decoupling limit in [135] and in the full theory in [150].

As already mentioned in two large extra dimensional models there is to be a maximal value ofthe cosmological constant that can be considered which is related to the six-dimensional Planckscale. Since that scale is in turn related to the effective mass of the graviton and since observationsset that scale to be relatively small, the model can only take care of a relatively small cosmologicalconstant. Nevertheless, it still provides a proof of principle on how to evade Weinberg’s no-gotheorem [484].

The extension of cascading gravity to more than two extra dimensions was considered in [149].It was shown in that case how the 3+ 1 brane remains flat for arbitrary values of the cosmologicalconstant on that brane (within the regime of validity of the weak-field approximation). See Figure 3for a picture on how the scalar potential adapts itself along the extra dimensions to accommodatefor a cosmological constant on the brane.

wy

z

4-brane

3-brane

5-brane

Figure 3: Seven-dimensional cascading scenario and solution for one the metric potential Φ on the (5+1)-dimensional brane in a 7-dimensional cascading gravity scenario with tension on the (3 + 1)-dimensionalbrane located at 𝑦 = 𝑧 = 0, in the case where 𝑀4

6 /𝑀35 = 𝑀5

7 /𝑀46 = 𝑚7. 𝑦 and 𝑧 represent the two extra

dimensions on the (5 + 1)-dimensional brane. Image reproduced with permission from [149], copyrightAPS.

9 Note that in DGP, one could also consider a smooth brane first and the results would remain unchanged.



42 Claudia de Rham

5 Deconstruction

As for DGP and its extensions, to get some insight on how to construct a four-dimensional theoryof single massive graviton, we can start with five-dimensional general relativity. This time, weconsider the extra dimension to be compactified and of finite size 𝑅, with periodic boundaryconditions. It is then natural to perform a Kaluza–Klein decomposition and to obtain a tower ofKaluza–Klein graviton mode in four dimensions. The zero mode is then massless and the highermodes are all massive with mass separation 𝑚 = 1/𝑅. Since the graviton mass is constant in thisformalism we omit the subscript 0 in the rest of this review.

Rather than starting directly with a Kaluza–Klein decomposition (discretization in Fourierspace), we perform instead a discretization in real space, known as “deconstruction” of five-dimensional gravity [24, 25, 170, 168, 28, 443, 340]. The deconstruction framework helps makingthe connection with massive gravity more explicit. However, we can also obtain multi-gravity outof it which is then completely equivalent to the Kaluza–Klein decomposition (after a non-linearfield redefinition).

The idea behind deconstruction is simply to ‘replace’ the continuous fifth dimension 𝑦 by aseries of 𝑁 sites 𝑦𝑗 separated by a distance ℓ = 𝑅/𝑁 . So that the five-dimensional metric isreplaced by a set of 𝑁 interacting metrics depending only on 𝑥.

In what follows, we review the procedure derived in [152] to recover four-dimensional ghost-freemassive gravity as well as bi- and multi-gravity out of five-dimensional GR. The procedure works inany dimensions and we only focus to deconstructing five-dimensional GR for sake of concreteness.

5.1 Formalism

5.1.1 Metric versus Einstein–Cartan formulation of GR

Before going further, let us first describe five-dimensional general relativity in its Einstein–Cartanformulation, where we introduce a set of vielbein 𝑒a𝐴, so that the relation between the metric andthe vielbein is simply,

𝑔𝐴𝐵(𝑥, 𝑦) = 𝑒a𝐴(𝑥, 𝑦)𝑒b

𝐵(𝑥, 𝑦)𝜂ab , (5.1)

where, as mentioned previously, the capital Latin letters label five-dimensional spacetime indices,while letters a, b, c · · · label five-dimensional Lorentz indices.

Under the torsionless condition, d𝑒+𝜔∧𝑒 = 0, the antisymmetric spin connection 𝜔, is uniquelydetermined in terms of the vielbeins

𝜔ab

𝐴 =1

2𝑒c𝐴(𝑂

ab

c−𝑂 ab

c−𝑂b a

c) , (5.2)

with 𝑂ab

c= 2𝑒a𝐴𝑒b𝐵 𝜕[𝐴𝑒𝐵]c. In the Einstein–Cartan formulation of GR, we introduce a 2-form

Riemann curvature,ℛab = d𝜔ab + 𝜔a

c∧ 𝜔cb , (5.3)

and up to boundary terms, the Einstein–Hilbert action is then given in the respective metric andthe vielbein languages by (here in five dimensions for definiteness),

𝑆(5)EH =

𝑀35

2

∫d4𝑥 d𝑦

√−𝑔 𝑅(5)[𝑔] (5.4)

=𝑀3

5

2× 3!

∫𝜀abcdeℛab ∧ 𝑒c ∧ 𝑒d ∧ 𝑒e , (5.5)

where 𝑅(5)[𝑔] is the scalar curvature built out of the five-dimensional metric 𝑔𝛼𝛽 and 𝑀5 is thefive-dimensional Planck scale.



Massive Gravity 43

The counting of the degrees of freedom in both languages is of course equivalent and goes asfollows: In 𝑑-spacetime dimensions, the metric has 𝑑(𝑑+1)/2 independent components. Covarianceremoves 2𝑑 of them,10 which leads to 𝒩𝑑 = 𝑑(𝑑 − 3)/2 independent degrees of freedom. In four-dimensions, we recover the usual 𝒩4 = 2 independent polarizations for gravitational waves. Infive-dimensions, this leads to 𝒩5 = 5 degrees of freedom which is the same number of degrees offreedom as a massive spin-2 field in four dimensions. This is as expect from the Kaluza–Kleinphilosophy (massless bosons in 𝑑 + 1 dimensions have the same number of degrees of freedom asmassive bosons in 𝑑 dimensions – this counting does not directly apply to fermions).

In the Einstein–Cartan formulation, the counting goes as follows: The vielbein has 𝑑2 indepen-dent components. Covariance removes 2𝑑 of them, and the additional global Lorentz invarianceremoves an additional 𝑑(𝑑 − 1)/2, leading once again to a total of 𝒩𝑑 = 𝑑(𝑑 − 3)/2 independentdegrees of freedom.

In GR one usually considers the metric and the vielbein formulation as being fully equivalent.However, this perspective is true only in the bosonic sector. The limitations of the metric formu-lation becomes manifest when coupling gravity to fermions. For such couplings one requires thevielbein formulation of GR. For instance, in four spacetime dimensions, the covariant action for aDirac fermion 𝜓 at the quadratic order is given by (see Ref. [392]),

𝑆Dirac =

∫1

3!𝜀𝑎𝑏𝑐𝑑 𝑒

𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐[𝑖

2𝜓𝛾𝑑←→𝐷 𝜓 − 𝑚

4𝑒𝑑𝜓𝜓

], (5.6)

where the 𝛾𝑎’s are the Dirac matrices and 𝐷 represents the covariant derivative, 𝐷𝜓 = 𝑑𝜓 −18𝜔

𝑎𝑏[𝛾𝑎, 𝛾𝑏]𝜓.In the bosonic sector, one can convert the covariant action of bosonic fields (e.g., of scalar, vector

fields, etc. . . ) between the vielbein and the metric language without much confusion, however thisis not possible for the covariant Dirac action, or other half-spin fields. For these types of matterfields, the Einstein–Cartan Formulation of GR is more fundamental than its metric formulation.In doubt, one should always start with the vielbein formulation. This is especially important inthe case of deconstruction when a discretization in the metric language is not equivalent to adiscretization in the vielbein variables. The same holds for Kaluza–Klein decomposition, a pointwhich might have been under-appreciated in the past.

5.1.2 Gauge-fixing

The discretization process breaks covariance and so before staring this procedure it is wise to fixthe gauge (failure to do so leads to spurious degrees of freedom which then become ghost in thefour-dimensional description). We thus start in five spacetime dimensions by setting the gauge

𝐺𝐴𝐵(𝑥, 𝑦) d𝑥𝐴 d𝑥𝐵 = d𝑦2 + 𝑔𝜇𝜈(𝑥, 𝑦) d𝑥

𝜇 d𝑥𝜈 , (5.7)

meaning that the lapse is set to unity and the shift to zero. Notice that one could in principleonly set the lapse to unity and keep the shift present throughout the discretization. From a four-dimensional point of view, the shift will then ‘morally’ play the role of the Stuckelberg fields,however they do so only after a cumbersome field redefinition. So for sake of clarity and simplicity,in what follows we first gauge-fix the shift and then once the four-dimensional theory is obtainedto restore gauge invariance by use of the Stuckelberg trick presented previously.

10 The local gauge invariance associated with covariance leads to 𝑑 first class constraints which remove 2𝑑 degrees offreedom, albeit in phase space. For global symmetries such as Lorentz invariance, there is no first-class constraintsassociated with them, and that global symmetry only removes 𝑑(𝑑 − 1)/2 degrees of freedom. Technically, thecounting should be performed in phase space, but the results remains the same. See Section 7.1 for a more detailedreview on the counting of degrees of freedom.



44 Claudia de Rham

In vielbein language, we fix the five-dimensional coordinate system and use four Lorentz trans-formations to set

𝑒a =

(𝑒𝑎𝜇 d𝑥𝜇

d𝑦

), (5.8)

and use the remaining six Lorentz transformations to set

𝜔𝑎𝑏𝑦 = 𝑒𝜇[𝑎𝜕𝑦𝑒𝑏]𝜇 = 0 . (5.9)

In this gauge, the five-dimensional Einstein–Hilbert term (5.4), (5.5) is given by

𝑆(5)EH =

𝑀35

2

∫d4𝑥 d𝑦

√−𝑔

(𝑅[𝑔] + [𝐾]2 − [𝐾2]

)(5.10)

=𝑀3

5

4

∫ (𝜀𝑎𝑏𝑐𝑑𝑅

𝑎𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 −𝐾𝑎 ∧𝐾𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 (5.11)

+2𝐾𝑎 ∧ 𝜕𝑦𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑)∧ d𝑦 ,

where 𝑅[𝑔], is the four-dimensional curvature built out of the four-dimensional metric 𝑔𝜇𝜈 , 𝑅𝑎𝑏 is

the 2-form curvature built out of the four-dimensional vielbein 𝑒𝑎𝜇 and its associated connection

𝜔𝑎𝑏 = 𝜔𝑎𝑏𝜇 d𝑥𝜇, 𝑅𝑎𝑏 = d𝜔𝑎𝑏 + 𝜔𝑎𝑐 ∧ 𝜔𝑐𝑏, and 𝐾𝜇𝜈 = 𝑔𝜇𝛼𝐾𝛼𝜈 is the extrinsic curvature,

𝐾𝜇𝜈 =1

2𝜕𝑦𝑔𝜇𝜈 = 𝑒𝑎(𝜇𝜕𝑦𝑒

𝑏𝜈) 𝜂𝑎𝑏 (5.12)

𝐾𝑎𝜇 = 𝑒𝜈𝑎𝐾𝜇𝜈 . (5.13)

5.1.3 Discretization in the vielbein

One could in principle go ahead and perform the discretization directly at the level of the metricbut first this would not lead to a consistent truncated theory of massive gravity.11 As explainedpreviously, the vielbein is more fundamental than the metric itself, and in what follows we discretizethe theory keeping the vielbein as the fundamental object.

𝑦 →˓ 𝑦𝑗 (5.14)

𝑒𝑎𝜇(𝑥, 𝑦) →˓ 𝑒𝑗𝑎𝜇(𝑥) = 𝑒𝑎𝜇(𝑥, 𝑦𝑗) (5.15)

𝜕𝑦𝑒𝑎𝜇(𝑥, 𝑦) →˓ 𝑚𝑁

(𝑒𝑗+1

𝑎𝜇 − 𝑒𝑗𝑎𝜇

). (5.16)

The gauge choice (5.9) then implies

𝜔𝑎𝑏𝑦 = 𝑒𝜇[𝑎𝜕𝑦𝑒𝑏]𝜇 = 0 →˓ 𝑒𝑗+1

𝜇[𝑎𝑒𝑗𝑏]𝜇 = 0 , (5.17)

where the arrow →˓ represents the deconstruction of five-dimensional gravity. We have also intro-duced the ‘truncation scale’, 𝑚𝑁 = 𝑁𝑚 = ℓ−1 = 𝑁𝑅−1, i.e., the scale of the highest mode inthe discretized theory. After discretization, we see the Deser–van Nieuwenhuizen [187] conditionappearing in Eq. (5.17), which corresponds to the symmetric vielbein condition. This is a sufficientcondition to allow for a formulation back into the metric language [410, 314, 172]. Note, however,that as mentioned in [152], we have not assumed that this symmetric vielbein condition was true,we simply derived it from the discretization procedure in the five-dimensional gauge choice 𝜔𝑎𝑏𝑦 = 0.

In terms of the extrinsic curvature, this implies

𝐾𝑎𝜇 →˓ 𝑚𝑁

(𝑒𝑗+1

𝑎𝜇 − 𝑒𝑗𝑎𝜇

). (5.18)

11 Discretizing at the level of the metric leads to a mass term similar to (2.83) which as we have seen contains aBD ghost.



Massive Gravity 45

This can be written back in the metric language as follows

𝑔𝜇𝜈(𝑥, 𝑦) →˓ 𝑔𝑗,𝜇𝜈(𝑥) = 𝑔𝜇𝜈(𝑥, 𝑦𝑗) (5.19)

𝐾𝜇𝜈 →˓ −𝑚𝑁𝒦𝜇𝜈 [𝑔𝑗 , 𝑔𝑗+1] ≡ −𝑚𝑁

(𝛿𝜇𝜈 −

(√𝑔−1𝑗 𝑔𝑗+1

)𝜇𝜈

), (5.20)

where the square root in the extrinsic curvature appears after converting back to the metric lan-guage. The square root exists as long as the metrics 𝑔𝑗 and 𝑔𝑗+1 have the same signature and𝑔−1𝑗 𝑔𝑗+1 has positive eigenvalues so if both metrics were diagonal the ‘time’ direction associatedwith each metric would be the same, which is a meaningful requirement.

From the metric language, we thus see that the discretization procedure amounts to convertingthe extrinsic curvature to an interaction between neighboring sites through the building block𝒦𝜇𝜈 [𝑔𝑗 , 𝑔𝑗+1].

5.2 Ghost-free massive gravity

5.2.1 Simplest discretization

In this subsection we focus on deriving a consistent theory of massive gravity from the discretizationprocedure (5.19, 5.20). For this, we consider a discretization with only two sites 𝑗 = 1, 2 and willonly be considered in the four-dimensional action induced on one site (say site 1), rather than thesum of both sites. This picture is analogous in spirit to a braneworld picture where we induce theaction at one point along the extra dimension. This picture gives the theory of a unique dynamicalmetric, expressed in terms of a reference metric which corresponds to the fixed metric on the othersite. We emphasize that this picture corresponds to a trick to build a consistent theory of massivegravity, and would otherwise be more artificial than its multi-gravity extension. However, as weshall see later, massive gravity can be seen as a perfectly consistent limit of multi (or bi-)gravitywhere the massless spin-2 field (and other fields in the multi-case) decouple and is thus perfectlyacceptable.

To simplify the notation for this two-site case, we write the vielbein on both sites as 𝑒1 = 𝑒,𝑒2 = 𝑓 , and similarly for the metrics 𝑔1,𝜇𝜈 = 𝑔𝜇𝜈 and 𝑔2,𝜇𝜈 = 𝑓𝜇𝜈 . Out of the five-dimensionalaction for GR, we obtain the theory of massive gravity in four dimensions, (on site 1),

𝑆(5)EH →˓ 𝑆

(4)mGR , (5.21)

with

𝑆(4)mGR =

𝑀2Pl

2

∫d4𝑥√−𝑔

(𝑅[𝑔] +𝑚2

([𝒦]2 − [𝒦2]

))(5.22)

=𝑀2

Pl

4

∫𝜀𝑎𝑏𝑐𝑑

(𝑅𝑎𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 +𝑚2𝒜𝑎𝑏𝑐𝑑(𝑒, 𝑓)

), (5.23)

with the mass term in the vielbein language

𝒜𝑎𝑏𝑐𝑑(𝑒, 𝑓) = (𝑓𝑎 − 𝑒𝑎) ∧ (𝑓𝑎 − 𝑒𝑎) ∧ 𝑒𝑐 ∧ 𝑒𝑑 , (5.24)

or the mass term building block in the metric language,

𝒦𝜇𝜈 = 𝛿𝜇𝜈 −(√

𝑔−1𝑓)𝜇𝜈. (5.25)

and we introduced the four-dimensional Planck scale, 𝑀2Pl =𝑀3

5

∫d𝑦, where in this case we limit

the integral about one site.



46 Claudia de Rham

The theory of massive gravity (5.22), or equivalently (5.23) is one special example of a ghost-freetheory of massive gravity (i.e., for which the BD ghost is absent). In terms of the ‘Stuckelbergized’tensor X introduced in Eq. (2.76), we see that

𝒦𝜇𝜈 = 𝛿𝜇𝜈 −(√

X)𝜇𝜈, (5.26)

or in other words,X𝜇𝜈 = 𝛿𝜇𝜈 − 2𝒦𝜇𝜈 +𝒦𝜇𝛼𝒦𝛼𝜈 , (5.27)

and the mass term can be written as

ℒmass = −𝑚2𝑀2

Pl

2

√−𝑔([𝒦2]− [𝒦]2

)(5.28)

= −𝑚2𝑀2

Pl

2

√−𝑔([(I−

√X)2]− [I−

√X]2). (5.29)

This also a generalization of the Fierz–Pauli mass term, albeit more complicated on first sight thanthe ones considered in (2.83) or (2.84), but as we shall see, a generalization of the Fierz–Pauli massterm which remains free of the BD ghost as is proven in depth in Section 7. We emphasize thatthe idea of the approach is not to give a proof of the absence of ghost (which is provided later) butrather to provide an intuitive argument of why the mass term takes its very peculiar structure.

5.2.2 Generalized mass term

This mass term is not the unique acceptable generalization of Fierz–Pauli gravity and by con-sidering more general discretization procedures we can generate the entire 2-parameter family ofacceptable potentials for gravity which will also be shown to be free of ghost in Section 7.

Rather than considering the straight-forward discretization 𝑒(𝑥, 𝑦) →˓ 𝑒𝑗(𝑥), we could considerthe average value on one site, pondered with arbitrary weight 𝑟,

𝑒(𝑥, 𝑦) →˓ 𝑟𝑒𝑗 + (1− 𝑟)𝑒𝑗+1 . (5.30)

The mass term at one site is then generalized to

𝐾𝑎 ∧𝐾𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 →˓ 𝑚2𝒜𝑎𝑏𝑐𝑑𝑟,𝑠 (𝑒𝑗 , 𝑒𝑗+1) , (5.31)

and the most general action for massive gravity with reference vielbein 𝑓 is thus12

𝑆mGR =𝑀2

Pl

4


(𝑅𝑎𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 +𝑚2𝒜𝑎𝑏𝑐𝑑𝑟,𝑠 (𝑒, 𝑓)

), (5.32)

with𝒜𝑎𝑏𝑐𝑑𝑟,𝑠 (𝑒, 𝑓) = (𝑓𝑎 − 𝑒𝑎) ∧ (𝑓 𝑏 − 𝑒𝑏) ∧ ((1− 𝑟)𝑒𝑐 + 𝑟𝑓 𝑐) ∧ ((1− 𝑠)𝑒𝑑 + 𝑠𝑓𝑑) ,

for any 𝑟, 𝑠 ∈ R.In particular for the two-site case, this generates the two-parameter family of mass terms

𝒜𝑎𝑏𝑐𝑑𝑟,𝑠 (𝑒, 𝑓) = 𝑐0 𝑒𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 + 𝑐1 𝑒

𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑓𝑑 (5.33)

+𝑐2 𝑒𝑎 ∧ 𝑒𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑 + 𝑐3 𝑒

𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑 + 𝑐4 𝑓𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑

≡ 𝒜1−𝑟,1−𝑠(𝑓, 𝑒) , (5.34)

12 This special fully non-linear and Lorentz invariant theory of massive gravity, which has been proven in allgenerality to be free of the BD ghost in [295, 296] has since then be dubbed ‘dRGT’ theory. To avoid any confusion,we thus also call this ghost-free theory of massive gravity, the dRGT theory.



Massive Gravity 47

with 𝑐0 = (1 − 𝑠)(1 − 𝑟), 𝑐1 = (−2 + 3𝑠 + 3𝑟 − 4𝑟𝑠), 𝑐2 = (1 − 3𝑠 − 3𝑟 + 6𝑟𝑠), 𝑐3 = (𝑟 + 𝑠 − 4𝑟𝑠)and 𝑐4 = 𝑟𝑠. This corresponds to the most general potential which, by construction, includes nocosmological constant nor tadpole. One can also always include a cosmological constant for suchmodels, which would naturally arise from a cosmological constant in the five-dimensional picture.

We see that in the vielbein language, the expression for the mass term is extremely naturaland simple. In fact this form was guessed at already for special cases in Ref. [410] and even earlierin [502]. However, the crucial analysis on the absence of ghosts and the reason for these termswas incorrect in both of these presentations. Subsequently, after the development of the consistentmetric formulation, the generic form of the mass terms was given in Refs. [95]13 and [314].

In the metric Language, this corresponds to the following Lagrangian for dRGT massive grav-ity [144], or its generalization to arbitrary reference metric [296]

ℒmGR =𝑀2

Pl

2

∫d4𝑥√−𝑔(𝑅+

𝑚2

2(ℒ2[𝒦] + 𝛼3ℒ3[𝒦] + 𝛼4ℒ4[𝒦])

), (5.35)

where the two parameters 𝛼3,4 are related to the two discretization parameters 𝑟, 𝑠 as

𝛼3 = 𝑟 + 𝑠, and 𝛼4 = 𝑟𝑠 , (5.36)

and for any tensor 𝑄, we define the scalar ℒ𝑛 symbolically as

ℒ𝑛[𝑄] = 𝜀𝜀𝑄𝑛 , (5.37)

for any 𝑛 = 0, · · · , 𝑑, where 𝑑 is the number of spacetime dimensions. 𝜀 is the Levi-Cevita antisym-metric symbol, so for instance in four dimensions, ℒ2[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼𝛽𝑄

𝜇′

𝜇 𝑄𝜈′

𝜈 = 2!([𝑄]2 − [𝑄2]),so we recover the mass term expressed in (5.28). Their explicit form is given in what follows inthe relations (6.11) – (6.13) or (6.16) – (6.18).

This procedure is easily generalizable to any number of dimensions, and massive gravity in 𝑑dimensions has (𝑑− 2)-free parameters which are related to the (𝑑− 2) discretization parameters.

5.3 Multi-gravity

In Section 5.2, we showed how to obtain massive gravity from considering the five-dimensionalEinstein–Hilbert action on one site.14 Instead in this section, we integrate over the whole of theextra dimension, which corresponds to summing over all the sites after discretization. Followingthe procedure of [152], we consider 𝑁 = 2𝑀 + 1 sites to start which leads to multi-gravity [314],and then focus on the two-site case leading to bi-gravity [293].

Starting with the five-dimensional action (5.12) and applying the discretization procedure (5.31)with 𝒜𝑎𝑏𝑐𝑑𝑟,𝑠 given in (5.33), we get

𝑆𝑁 mGR =𝑀2

4

4

𝑁∑𝑗=1


(𝑅𝑎𝑏[𝑒𝑗 ] ∧ 𝑒𝑐𝑗 ∧ 𝑒𝑑𝑗 +𝑚2

𝑁𝒜𝑎𝑏𝑐𝑑𝑟𝑗 ,𝑠𝑗 (𝑒𝑗 , 𝑒𝑗+1))

(5.38)

=𝑀2

4

2

𝑁∑𝑗=1

∫d4𝑥√−𝑔𝑗

(𝑅[𝑔𝑗 ] +

𝑚2𝑁

2

4∑𝑛=0

𝛼(𝑗)𝑛 ℒ𝑛(𝒦𝑗,𝑗+1)

),

13 The analysis performed in Ref. [95] was unfortunately erroneous, and the conclusions of that paper are thusincorrect.

14 In the previous section we obtained directly a theory of massive gravity, this should be seen as a trick to obtaina consistent theory of massive gravity. However, we shall see that we can take a decoupling limit of bi- (or evenmulti-)gravity so as to recover massive gravity and a decoupled massless spin-2 field. In this sense massive gravityis a perfectly consistent limit of bi-gravity.



48 Claudia de Rham

with 𝑀24 = 𝑀3

5𝑅 = 𝑀35 /𝑚, 𝛼

(𝑗)2 = −1/2, and in this deconstruction framework we obtain no

cosmological constant nor tadpole, 𝛼(𝑗)0 = 𝛼

(𝑗)1 = 0 at any site 𝑗, (but we keep them for generality).

In the mass Lagrangian, we use the shorthand notation 𝒦𝑗,𝑗+1 for the tensor 𝒦𝜇𝜈 [𝑔𝑗 , 𝑔𝑗+1]. Thisis a special case of multi-gravity presented in [314] (see also [417] for other ‘topologies’ in the waythe multiple gravitons interact), where each metric only interacts with two other metrics, i.e., with

its closest neighbors, leading to 2𝑁 -free parameters. For any fixed 𝑗, one has 𝛼(𝑗)3 = (𝑟𝑗 + 𝑠𝑗), and

𝛼(𝑗)4 = 𝑟𝑗𝑠𝑗 .To see the mass spectrum of this multi-gravity theory, we perform a Fourier decomposition,

which is what one would obtain (after a field redefinition) by performing a KK decompositionrather than a real space discretization. KK decomposition and deconstruction are thus perfectlyequivalent (after a non-linear – but benign15 – field redefinition). We define the discrete Fouriertransform of the vielbein variables,

𝑒𝑎𝜇,𝑛 =1√𝑁

𝑁∑𝑗=1

𝑒𝑎𝜇,𝑗𝑒𝑖 2𝜋𝑁 𝑗 , (5.39)

with the inverse map,

𝑒𝑎𝜇,𝑗 =1√𝑁

𝑀∑𝑛=−𝑀

𝑒𝑎𝜇,𝑛𝑒−𝑖 2𝜋𝑁 𝑛 . (5.40)

In terms of the Fourier transform variables, the multi-gravity action then reads at the linear level

ℒ =

𝑀∑𝑛=−𝑀

[(𝜕ℎ𝑛)(𝜕ℎ−𝑛) +𝑚2

𝑛ℎ𝑛ℎ−𝑛

]+ ℒint (5.41)

with 𝑀−1Pl ℎ𝜇𝜈,𝑛 = 𝑒𝑎𝜇,𝑛𝑒

𝑏𝜈,𝑛𝜂𝑎𝑏 − 𝜂𝜇𝜈 and 𝑀Pl represents the four-dimensional Planck scale, 𝑀Pl =

𝑀4/√𝑁 . The reality condition on the vielbein imposes 𝑒𝑛 = 𝑒*−𝑛 and similarly for ℎ𝑛. The mass

spectrum is then

𝑚𝑛 = 𝑚𝑁 sin( 𝑛𝑁

)≈ 𝑛𝑚 for 𝑛≪ 𝑁. (5.42)

The counting of the degrees of freedom in multi-gravity goes as follows: the theory contains2𝑀 massive spin-2 fields with five degrees of freedom each and one massless spin-2 field with twodegrees of freedom, corresponding to a total of 10𝑀+2 degrees of freedom. In the continuum limit,we also need to account for the zero mode of the lapse and the shift which have been gauged fixedin five dimensions (see Ref. [443] for a nice discussion of this point). This leads to three additionaldegrees of freedom, summing up to a total of 5𝑁 degrees of freedom of the four coordinates 𝑥𝑎.

5.4 Bi-gravity

Let us end this section with the special case of bi-gravity. Bi-gravity can also be derived fromthe deconstruction paradigm, just as massive gravity and multi-gravity, but the idea has beeninvestigated for many years (see for instance [436, 324]). Like massive gravity, bi-gravity was for along time thought to host a BD ghost parasite, but a ghost-free realization was recently proposedby Hassan and Rosen [293] and bi-gravity is thus experiencing a revived amount of interested.This extensions is nothing other than the ghost-free massive gravity Lagrangian for a dynamicalreference metric with the addition of an Einstein–Hilbert term for the now dynamical referencemetric.

15 The field redefinition is local so no new degrees of freedom or other surprises hide in that field redefinition.



Massive Gravity 49

Bi-gravity from deconstruction

Let us consider a two-site discretization with periodic boundary conditions, 𝑗 = 1, 2, 3 with quan-tities at the site 𝑗 = 3 being identified with that at the site 𝑗 = 1. Similarly, as in Section 5.2we denote by 𝑔𝜇𝜈 = 𝑒𝑎𝜇𝑒

𝑏𝜈𝜂𝑎𝑏 and by 𝑓𝜇𝜈 =ê𝑎𝜇 ê𝑏𝜈𝜂𝑎𝑏 the metrics and vielbeins at the respective

locations 𝑦1 and 𝑦2.Then applying the discretization procedure highlighted in Eqs. (5.14, 5.15, 5.18, 5.19 and 5.20)

and summing over the extra dimension, we obtain the bi-gravity action

𝑆bi−gravity =𝑀2

Pl

2

∫d4𝑥√−𝑔𝑅[𝑔] +

𝑀2𝑓

2

∫d4𝑥√−𝑓𝑅[𝑓 ] (5.43)

+𝑀2

Pl𝑚2

4

∫d4𝑥√−𝑔

4∑𝑛=0

𝛼𝑛ℒ𝑛[𝒦[𝑔, 𝑓 ]] ,

where 𝒦[𝑔, 𝑓 ] is given in (5.25) and we use the notation 𝑀𝑔 =𝑀Pl. We can equivalently well writethe mass terms in terms of 𝒦[𝑓, 𝑔] rather than 𝒦[𝑔, 𝑓 ] as performed in (6.21).

Notice that the most naive discretization procedure would lead to 𝑀𝑔 =𝑀Pl =𝑀𝑓 , but thesecan be generalized either ‘by hand’ by changing the weight of each site during the discretization,or by considering a non-trivial configuration along the extra dimension (for instance warping alongthe extra dimension16), or most simply by performing a conformal rescaling of the metric at eachsite.

Here, ℒ0[𝒦[𝑔, 𝑓 ]] corresponds to a cosmological constant for the metric 𝑔𝜇𝜈 and the special

combination∑4𝑛=0(−1)𝑛𝐶𝑛4 ℒ𝑛[𝒦[𝑔, 𝑓 ]], where the 𝐶𝑚𝑛 are the binomial coefficients is the cosmo-

logical constant for the metric 𝑓𝜇𝜈 , so only ℒ2,3,4 correspond to genuine interactions between thetwo metrics.

In the deconstruction framework, we naturally obtain 𝛼2 = 1 and no tadpole nor cosmologicalconstant for either metrics.

Mass eigenstates

In this formulation of bi-gravity, both metrics 𝑔 and 𝑓 carry a superposition of the massless andthe massive spin-2 field. As already emphasize the notion of mass (and of spin) only makes sensefor a field living in Minkowski, and so to analyze the mass spectrum, we expand both metricsabout flat spacetime,

𝑔𝜇𝜈 = 𝜂𝜇𝜈 +1

𝑀Pl𝛿𝑔𝜇𝜈 (5.44)

𝑓𝜇𝜈 = 𝜂𝜇𝜈 +1

𝑀𝑓𝛿𝑓𝜇𝜈 . (5.45)

The general mass spectrum about different backgrounds is richer and provided in [300]. Here weonly focus on a background which preserves Lorentz invariance (in principle we could also includeother maximally symmetric backgrounds which hae the same amount of symmetry as Minkowski).

Working about Minkowski, then to quadratic order in ℎ, the action for bi-gravity reads (for𝛼0 = 𝛼1 = 0 and 𝛼2 = −1/2),

𝑆(2)bi−gravity =

∫d4𝑥[− 1

4𝛿𝑔𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝛿𝑔𝛼𝛽 −

1

4𝛿𝑓𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝛿𝑓𝛼𝛽 −

1

8𝑚2

eff

(ℎ2𝜇𝜈 − ℎ2

) ], (5.46)

16 See Refs. [169, 434, 242, 340] for additional work on deconstruction in five-dimensional AdS, and how thistackles the strong coupling issue.



50 Claudia de Rham

where all indices are raised and lowered with respect to the flat Minkowski metric and the Lich-nerowicz operator ℰ𝛼𝛽𝜇𝜈 was defined in (2.37). We see appearing the Fierz–Pauli mass term com-bination ℎ2𝜇𝜈 − ℎ2 introduced in (2.44) for the massive field with the effective mass 𝑀eff definedas [293]

𝑀2eff =

(𝑀−2

Pl +𝑀−2𝑓

)−1

(5.47)

𝑚2eff = 𝑚2𝑀

2Pl

𝑀2eff

. (5.48)

The massive field ℎ is given by

ℎ𝜇𝜈 =𝑀eff

(1

𝑀Pl𝛿𝑔𝜇𝜈 −

1

𝑀𝑓𝛿𝑓𝜇𝜈

)=𝑀eff (𝑔𝜇𝜈 − 𝑓𝜇𝜈) , (5.49)

while the other combination represents the massless field ℓ𝜇𝜈 ,

ℓ𝜇𝜈 =𝑀eff

(1

𝑀𝑓𝛿𝑔𝜇𝜈 +

1

𝑀Pl𝛿𝑓𝜇𝜈

), (5.50)

so that in terms of the light and heavy spin-2 fields (or more precisely in terms of the two masseigenstates ℎ and ℓ), the quadratic action for bi-gravity reproduces that of a massless spin-2 fieldℓ and a Fierz–Pauli massive spin-2 field ℎ with mass 𝑚eff ,

𝑆(2)bi−gravity =

∫d4𝑥[−1

4ℎ𝜇𝜈[ℰ𝛼𝛽𝜇𝜈 +

1

2𝑚2

eff

(𝛿𝛼𝜇𝛿

𝛽𝜈 − 𝜂𝛼𝛽𝜂𝜇𝜈

) ]ℎ𝛼𝛽 (5.51)

−1

4ℓ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℓ𝛼𝛽

].

As explained in [293], in the case where there is a large Hierarchy between the two Planck scales𝑀Pl and 𝑀𝑓 , the massive particles is always the one that enters at the lower Planck mass andthe massless one the one that has a large Planck scale. For instance if 𝑀𝑓 ≫ 𝑀Pl, the masslessparticle is mainly given by 𝛿𝑓𝜇𝜈 and the massive one mainly by 𝛿𝑔𝜇𝜈 . This means that in the limit𝑀𝑓 →∞ while keeping𝑀Pl fixed, we recover the theory of a massive gravity and a fully decoupledmassless graviton as will be explained in Section 8.2.

5.5 Coupling to matter

So far we have only focus on an empty five-dimensional bulk with no matter. It is natural,though, to consider matter fields living in five dimensions, 𝜒(𝑥, 𝑦) with Lagrangian (in the gaugechoice (5.7))

ℒmatter =√−𝑔(−1

2(𝜕𝜇𝜒)

2 − 1

2(𝜕𝑦𝜒)

2 − 𝑉 (𝜒)

), (5.52)

in addition to arbitrary potentials (we focus on the case of a scalar field for simplicity, but thesame philosophy can be applied to higher-spin species be it bosons or fermions). Then applyingthe same discretization scheme used for gravity, every matter field then comes in 𝑁 copies

𝜒(𝑥, 𝑦) →˓ 𝜒(𝑗)(𝑥) = 𝜒(𝑥, 𝑦𝑗) , (5.53)

for 𝑗 = 1, · · · , 𝑁 and each field 𝜒(𝑗) is coupled to the associated vielbein 𝑒(𝑗) or metric 𝑔(𝑗)𝜇𝜈 =

𝑒(𝑗) 𝑎𝜇 𝑒

(𝑗) 𝑏𝜈 𝜂𝑎𝑏 at the same site. In the discretization procedure, the gradient along the extra dimen-

sion yields a mixing (interaction) between fields located on neighboring sites,∫d𝑦 (𝜕𝑦𝜒)

2 →˓ 𝑅𝑁∑𝑗=1

𝑚2(𝜒(𝑗+1)(𝑥)− 𝜒(𝑗)(𝑥))2 , (5.54)



Massive Gravity 51

(assuming again periodic boundary conditions, 𝜒(𝑁+1) = 𝜒(1)). The discretization procedure couldbe also performed using a more complicated definition of the derivative along 𝑦 involving morethan two sites, which leads to further interactions between the different fields.

In the two-sight derivative formulation, the action for matter is then

𝑆matter →˓1

𝑚

∫d4𝑥

∑𝑗

√−𝑔(𝑗)

(− 1

2𝑔(𝑗)𝜇𝜈𝜕𝜇𝜒

(𝑗)𝜕𝜈𝜒(𝑗) (5.55)

−1

2𝑚2(𝜒(𝑗+1) − 𝜒(𝑗))2 − 𝑉 (𝜒(𝑗))

).

The coupling to gauge fields or fermions can be derived in the same way, and the vielbein formalismmakes it natural to extend the action (5.6) to five dimensions and applying the discretizationprocedure. Interestingly, in the case of fermions, the fields 𝜓(𝑗) and 𝜓(𝑗+1) would not directlycouple to one another, but they would couple to both the vielbein 𝑒(𝑗) at the same site and theone 𝑒(𝑗−1) on the neighboring site.

Notice, however, that the current full proofs for the absence of the BD ghost do not includesuch couplings between matter fields living on different metrics (or vielbeins), nor matter fieldscoupling directly to more than one metric (vielbein).

5.6 No new kinetic interactions

In GR, diffeomorphism invariance uniquely fixes the kinetic term to be the Einstein–Hilbert one

ℒEH =√−𝑔𝑅 , (5.56)

(see, for instance, Refs. [287, 483, 175, 225, 76] for the uniqueness of GR for the theory of a masslessspin-2 field).

In more than four dimensions, the GR action can be supplemented by additional Lovelockinvariants [383] which respect diffeomorphism invariance and are expressed in terms of higherpowers of the Riemann curvature but lead to second order equations of motion. In four dimensionsthere is only one non-trivial additional Lovelock invariant corresponding the Gauss–Bonnet termbut it is topological and thus does not affect the theory, unless other degrees of freedom such as ascalar field is included.

So, when dealing with the theory of a single massless spin-2 field in four dimensions the onlyallowed kinetic term is the well-known Einstein–Hilbert one. Now when it comes to the theoryof a massive spin-2 field, diffeomorphism invariance is broken and so in addition to the allowedpotential terms described in (6.9) – (6.13), one could consider other kinetic terms which breakdiffeomorphism.

This possibility was explored in Refs. [231, 310, 230] where it was shown that in four dimensions,

the following derivative interaction ℒ(der)3 is ghost-free at leading order (i.e., there is no higher

derivatives for the Stuckelberg fields when introducing the Stuckelberg fields associated with lineardiffeomorphism),

ℒ(der)3 = 𝜀𝜇𝜈𝜌𝜎𝜀𝜇

′𝜈′𝜌′𝜎′ℎ𝜎𝜎′𝜕𝜌ℎ𝜇𝜇′𝜕𝜌′ℎ𝜈𝜈′ . (5.57)

So this new derivative interaction would be allowed for a theory of a massive spin-2 field whichdoes not couple to matter. Note that this interaction can only be considered if the spin-2 field ismassive in the first place, so this interaction can only be present if the Fierz–Pauli mass term (2.44)is already present in the theory.

Now let us turn to a theory of gravity. In that case, we have seen that the coupling to matterforces linear diffeomorphisms to be extended to fully non-linear diffeomorphism. So to be viablein a theory of massive gravity, the derivative interaction (5.57) should enjoy a ghost-free non-linear



52 Claudia de Rham

completion (the absence of ghost non-linearly can be checked for instance by restoring non-lineardiffeomorphism using the non-linear Stuckelberg decomposition (2.80) in terms of the helicity-1and -0 modes given in (2.46), or by performing an ADM analysis as will be performed for the mass

term in Section 7.) It is easy to check that by itself ℒ(der)3 has a ghost at quartic order and so other

non-linear interactions should be included for this term to have any chance of being ghost-free.

Within the deconstruction paradigm, the non-linear completion of ℒ(der)3 could have a natural

interpretation as arising from the five-dimensional Gauss–Bonnet term after discretization. Ex-ploring the avenue would indeed lead to a new kinetic interaction of the form

√−𝑔𝒦𝜇𝜈𝒦𝛼𝛽*𝑅𝜇𝜈𝛼𝛽 ,

where *𝑅 is the dual Riemann tensor [339, 153]. However, a simple ADM analysis shows that such aterm propagates more than five degrees of freedom and thus has an Ostrogradsky ghost (similarlyas the BD ghost). As a result this new kinetic interaction (5.57) does not have a natural realizationfrom a five-dimensional point of view (at least in its metric formulation, see Ref. [153] for moredetails.)

We can push the analysis even further and show that no matter what the higher order inter-

actions are, as soon as ℒ(der)3 is present it will always lead to a ghost and so such an interaction is

never acceptable [153].As a result, the Einstein–Hilbert kinetic term is the only allowed kinetic term in Lorentz-

invariant (massive) gravity.This result shows how special and unique the Einstein–Hilbert term is. Even without imposing

diffeomorphism invariance, the stability of the theory fixes the kinetic term to be nothing else thanthe Einstein–Hilbert term and thus forces diffeomorphism invariance at the level of the kinetic term.Even without requiring coordinate transformation invariance, the Riemann curvature remains thebuilding block of the kinetic structure of the theory, just as in GR.

Before summarizing the derivation of massive gravity from higher dimensional deconstruc-tion /Kaluza–Klein decomposition, we briefly comment on other ‘apparent’ modifications of thekinetic structure like in 𝑓(𝑅) – gravity (see for instance Refs. [89, 354, 46] for 𝑓(𝑅) massive gravityand their implications to cosmology).

Such kinetic terms a la 𝑓(𝑅) are also possible without a mass term for the graviton. In thatcase diffeomorphism invariance allows us to perform a change of frame. In the Einstein-frame 𝑓(𝑅)gravity is seen to correspond to a theory of gravity with a scalar field, and the same result will holdin 𝑓(𝑅) massive gravity (in that case the scalar field couples non-trivially to the Stuckelberg fields).As a result 𝑓(𝑅) is not a genuine modification of the kinetic term but rather a standard Einstein–Hilbert term and the addition of a new scalar degree of freedom which not a degree of freedom ofthe graviton but rather an independent scalar degree of freedom which couples non-minimally tomatter (see Ref. [128] for a review on 𝑓(𝑅)-gravity.)



Massive Gravity 53

Part II

Ghost-free Massive Gravity

6 Massive, Bi- and Multi-Gravity Formulation: A Summary

The previous ‘deconstruction’ framework gave a intuitive argument for the emergence of a potentialof the form (6.3) (or (6.1) in the vielbein language) and its bi- and multi-metric generalizations.In deconstruction or Kaluza–Klein decomposition a certain type of interaction arises naturally andwe have seen that the whole spectrum of allowed potentials (or interactions) could be generatedby extending the deconstruction procedure to a more general notion of derivative or by involvingthe mixing of more sites in the definition of the derivative along the extra dimensions. We heresummarize the most general formulation for the theories of massive gravity about a generic referencemetric, bi-gravity and multi-gravity and provide a dictionary between the different languages usedin the literature.

The general action for ghost-free (or dRGT) massive gravity [144] in the vielbein languageis [95, 314] (see however Footnote 13 with respect to Ref. [95], see also Refs. [502, 410] for earlierwork)

𝑆mGR =𝑀2

Pl

4


𝑎𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 +𝑚2ℒ(mass)(𝑒, 𝑓)), (6.1)

with

ℒ(mass)(𝑒, 𝑓) = 𝜀𝑎𝑏𝑐𝑑

[𝑐0 𝑒

𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 + 𝑐1 𝑒𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑓𝑑

+𝑐2 𝑒𝑎 ∧ 𝑒𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑 + 𝑐3 𝑒

𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑

+𝑐4 𝑓𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑

],

(6.2)

or in the metric language [144],

𝑆mGR =𝑀2

Pl

2

∫d4𝑥√−𝑔

(𝑅+

𝑚2

2

4∑𝑛=0

𝛼𝑛ℒ𝑛[𝒦[𝑔, 𝑓 ]]

). (6.3)

In what follows we will use the notation for the overall potential of massive gravity

𝒰 = −𝑀2Pl

4

√−𝑔

4∑𝑛=0

𝛼𝑛ℒ𝑛[𝒦[𝑔, 𝑓 ]] = −ℒ(mass)(𝑒, 𝑓) , (6.4)

so thatℒmGR =𝑀2

PlℒGR[𝑔]−𝑚2 𝒰 [𝑔, 𝑓 ] , (6.5)

where ℒGR[𝑔] is the standard GR Einstein–Hilbert Lagrangian for the dynamical metric 𝑔𝜇𝜈 and𝑓𝜇𝜈 is the reference metric and for bi-gravity,

ℒbi−gravity =𝑀2PlℒGR[𝑔] +𝑀2

𝑓ℒGR[𝑓 ]−𝑚2 𝒰 [𝑔, 𝑓 ] , (6.6)

where both 𝑔𝜇𝜈 and 𝑓𝜇𝜈 are then dynamical metrics.Both massive gravity and bi-gravity break one copy of diff invariance and so the Stuckelberg

fields can be introduced in exactly the same way in both cases 𝒰 [𝑔, 𝑓 ] → 𝒰 [𝑔, 𝑓 ] where the



54 Claudia de Rham

Stuckelbergized metric 𝑓𝜇𝜈 was introduced in (2.75) (or alternatively 𝒰 [𝑔, 𝑓 ] → 𝒰 [𝑔, 𝑓 ]). Thusbi-gravity is by no means an alternative to introducing the Stuckelberg fields as is sometimesstated.

In these formulations, ℒ0 (or the term proportional to 𝑐0) correspond to a cosmological constant,ℒ1 to a tadpole, ℒ2 to the mass term and ℒ3,4 to allowed higher order interactions. The presenceof the tadpole ℒ1 would imply a non-zero vev. The presence of the potentials ℒ3,4 without ℒ2

would lead to infinitely strongly coupled degrees of freedom and would thus be pathological. Werecall that 𝒦[𝑔, 𝑓 ] is given in terms of the metrics 𝑔 and 𝑓 as

𝒦𝜇𝜈 [𝑔, 𝑓 ] = 𝛿𝜇𝜈 −(√

𝑔−1𝑓)𝜇𝜈, (6.7)

and the Lagrangians ℒ𝑛 are defined as follows in arbitrary dimensions 𝑑 [144]

ℒ𝑛[𝑄] = −(𝑑− 𝑛)!𝑛∑

𝑚=1

(−1)𝑚 (𝑛− 1)!

(𝑛−𝑚)!(𝑑− 𝑛+𝑚)![𝑄𝑚]ℒ(𝑛−𝑚)

𝑛 [𝑄] , (6.8)

with ℒ0[𝑄] = 𝑑! and ℒ1[𝑄] = (𝑑− 1)![𝑄] or equivalently in four dimensions [292]

ℒ0[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇𝜈𝛼𝛽 (6.9)

ℒ1[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈𝛼𝛽 𝑄𝜇′

𝜇 (6.10)

ℒ2[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼𝛽𝑄𝜇′

𝜈 𝑄𝜈′

𝜈 (6.11)

ℒ3[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼′𝛽𝑄𝜇′

𝜈 𝑄𝜈′

𝜈 𝑄𝛼′

𝛼 (6.12)

ℒ4[𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼′𝛽′𝑄𝜇′

𝜈 𝑄𝜈′

𝜈 𝑄𝛼′

𝛼 𝑄𝛽′

𝛽 . (6.13)

We have introduced the constant ℒ0 (ℒ0 = 4! and√−𝑔ℒ0 is nothing other than the cosmological

constant) and the tadpole ℒ1 for completeness. Notice however that not all these five Lagrangiansare independent and the tadpole can always be re-expressed in terms of a cosmological constantand the other potential terms.

Alternatively, we may express these scalars as follows [144]

ℒ0[𝑄] = 4! (6.14)

ℒ1[𝑄] = 3! [𝑄] (6.15)

ℒ2[𝑄] = 2!([𝑄]2 − [𝑄2]) (6.16)

ℒ3[𝑄] = ([𝑄]3 − 3[𝑄][𝑄2] + 2[𝑄3]) (6.17)

ℒ4[𝑄] = ([𝑄]4 − 6[𝑄]2[𝑄2] + 3[𝑄2]2 + 8[𝑄][𝑄3]− 6[𝑄4]) . (6.18)

These are easily generalizable to any number of dimensions, and in 𝑑 dimensions we find 𝑑 suchindependent scalars.

The multi-gravity action is a generalization to multiple interacting spin-2 fields with the sameform for the interactions, and bi-gravity is the special case of two metrics (𝑁 = 2), [314]

𝑆𝑁 =𝑀2

Pl

4

𝑁∑𝑗=1


𝑎𝑏[𝑒𝑗 ] ∧ 𝑒𝑐𝑗 ∧ 𝑒𝑑𝑗 +𝑚2𝑁ℒ(mass)(𝑒𝑗 , 𝑒𝑗+1)

), (6.19)

or

𝑆𝑁 =𝑀2

Pl

2

𝑁∑𝑗=1

∫d4𝑥√−𝑔𝑗

(𝑅[𝑔𝑗 ] +

𝑚2𝑁

2

4∑𝑛=0

𝛼(𝑗)𝑛 ℒ𝑛[𝒦[𝑔𝑗 , 𝑔𝑗+1]]

). (6.20)



Massive Gravity 55

Inverse argument

We could have written this set of interactions in terms of 𝒦[𝑓, 𝑔] rather than 𝒦[𝑔, 𝑓 ],

𝒰 =𝑀2

Pl𝑚2

4

∫d4𝑥√−𝑔

4∑𝑛=0


=𝑀2

Pl𝑚2

4

∫d4𝑥√−𝑓

4∑𝑛=0

��𝑛ℒ𝑛[𝒦[𝑓, 𝑔]] , (6.21)

with ⎛⎜⎜⎜⎜⎝��0

��1

��2

��3

��4

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝1 0 0 0 0−4 −1 0 0 06 3 1 0 0−4 −3 −2 −1 01 1 1 1 1

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝛼0

𝛼1

𝛼2

𝛼3

𝛼4

⎞⎟⎟⎟⎟⎠ . (6.22)

Interestingly, the absence of tadpole and cosmological constant for say the metric 𝑔 implies 𝛼0 =𝛼1 = 0 which in turn implies the absence of tadpole and cosmological constant for the other metric𝑓 , ��0 = ��1 = 0, and thus ��2 = 𝛼2 = 1.

Alternative variables

Alternatively, another fully equivalent convention has also been used in the literature [292] in termsof X𝜇𝜈 = 𝑔𝜇𝛼𝑓𝛼𝜈 defined in (2.76),

𝒰 = −𝑀2Pl

4

√−𝑔

4∑𝑛=0

𝛽𝑛𝑛!ℒ𝑛[√X] , (6.23)

which is equivalent to (6.4) with ℒ0 = 4! and⎛⎜⎜⎜⎜⎝𝛽0𝛽1𝛽2𝛽3𝛽4

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝1 1 1 1 10 −1 −2 −3 −40 0 2 6 120 0 0 −6 −240 0 0 0 24

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝛼0

𝛼1

𝛼2

𝛼3

𝛼4

⎞⎟⎟⎟⎟⎠ , (6.24)

or the inverse relation, ⎛⎜⎜⎜⎜⎝𝛼0

𝛼1

𝛼2

𝛼3

𝛼4

⎞⎟⎟⎟⎟⎠ =1

24

⎛⎜⎜⎜⎜⎝24 24 12 4 10 −24 −24 −12 −40 0 12 12 60 0 0 −4 −40 0 0 0 1

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝛽0𝛽1𝛽2𝛽3𝛽4

⎞⎟⎟⎟⎟⎠ , (6.25)

so that in order to avoid a tadpole and a cosmological constant we need to set for instance 𝛽4 =−(24𝛽0 + 24𝛽1 + 12𝛽2 + 4𝛽3) and 𝛽3 = −6(4𝛽0 + 3𝛽1 + 𝛽2).

Expansion about the reference metric

In the vielbein language the mass term is extremely simple, as can be seen in Eq. (6.1) with 𝒜defined in (2.60). Back to the metric language, this means that the mass term takes a remarkably



56 Claudia de Rham

simple form when writing the dynamical metric 𝑔𝜇𝜈 in terms of the reference metric 𝑓𝜇𝜈 and a

difference ℎ𝜇𝜈 = 2ℎ𝜇𝜈 + ℎ2𝜇𝜈 as

𝑔𝜇𝜈 = 𝑓𝜇𝜈 + 2ℎ𝜇𝜈 + ℎ𝜇𝛼ℎ𝜈𝛽𝑓𝛼𝛽 , (6.26)

where 𝑓𝛼𝛽 = (𝑓−1)𝛼𝛽 . The mass terms is then expressed as

𝒰 = −𝑀2Pl

4

√−𝑓

4∑𝑛=0

𝜅𝑛ℒ𝑛[𝑓𝜇𝛼ℎ𝛼𝜈 ] , (6.27)

where the ℒ𝑛 have the same expression as the ℒ𝑛 in (6.9) – (6.13) so ℒ𝑛 is genuinely 𝑛th orderin ℎ𝜇𝜈 . The expression (6.27) is thus at most quartic order in ℎ𝜇𝜈 but is valid to all orders inℎ𝜇𝜈 , (there is no assumption that ℎ𝜇𝜈 be small). In other words, the mass term (6.27) is notan expansion in ℎ𝜇𝜈 truncated to a finite (quartic) order, but rather a fully equivalent way torewrite the mass Lagrangian in terms of the variable ℎ𝜇𝜈 rather than 𝑔𝜇𝜈 . Of course the kineticterm is intrinsically non-linear and includes a infinite expansion in ℎ𝜇𝜈 . A generalization of suchparameterizations are provided in [300].

The relation between the coefficients 𝜅𝑛 and 𝛼𝑛 is given by⎛⎜⎜⎜⎜⎝𝜅0𝜅1𝜅2𝜅3𝜅4

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝1 0 0 0 04 1 0 0 06 3 1 0 04 3 3 1 01 1 1 1 1

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝛼0

𝛼1

𝛼2

𝛼3

𝛼4

⎞⎟⎟⎟⎟⎠ . (6.28)

The quadratic expansion about a background different from the reference metric was derivedin Ref. [278]. Notice however that even though the mass term may not appear as having an exactFierz–Pauli structure as shown in [278], it still has the correct structure to avoid any BD ghost,about any background [295, 294, 300, 297].

7 Evading the BD Ghost in Massive Gravity

The deconstruction framework gave an intuitive approach on how to construct a theory of massivegravity or multiple interacting ‘gravitons’. This lead to the ghost-free dRGT theory of massivegravity and its bi- and multi-gravity extensions in a natural way. However, these developmentswere only possible a posteriori.

The deconstruction framework was proposed earlier (see Refs. [24, 25, 168, 28, 443, 168, 170])directly in the metric language and despite starting from a perfectly healthy five-dimensional theoryof GR, the discretization in the metric language leads to the standard BD issue (this also holds ina KK decomposition when truncating the KK tower at some finite energy scale). Knowing thatmassive gravity (or multi-gravity) can be naturally derived from a healthy five-dimensional theoryof GR is thus not a sufficient argument for the absence of the BD ghost, and a great amountof effort was devoted to that proof, which is known by now a multitude of different forms andlanguages.

Within this review, one cannot make justice to all the independent proofs that have beenformulated by now in the literature. We thus focus on a few of them – the Hamiltonian analysis inthe ADM language – as well as the analysis in the Stuckelberg language. One of the proofs in thevielbein formalism will be used in the multi-gravity case, and thus we do not emphasize that proofin the context of massive gravity, although it is perfectly applicable (and actually very elegant) inthat case. Finally, after deriving the decoupling limit in Section 8.3, we also briefly review how itcan be used to prove the absence of ghost more generically.



Massive Gravity 57

We note that even though the original argument on how the BD ghost could be circumventedin the full nonlinear theory was presented in [137] and [144], the absence of BD ghost in “ghost-freemassive gravity” or dRGT has been the subject of many discussions [12, 13, 345, 342, 95, 341, 344,96] (see also [350, 351, 349, 348, 352] for related discussions in bi-gravity). By now the confusionhas been clarified, and see for instance [295, 294, 400, 346, 343, 297, 15, 259] for thorough proofsaddressing all the issues raised in the previous literature. (See also [347] for the proof of the absenceof ghosts in other closely related models).

7.1 ADM formulation

7.1.1 ADM formalism for GR

Before going onto the subtleties associated with massive gravity, let us briefly summarize how thecounting of the number of degrees of freedom can be performed in the ADM language using theHamiltonian for GR. Using an ADM decomposition (where this time, we single out the time, ratherthan the extra dimension as was performed in Part I),

d𝑠2 = −𝑁2 d𝑡2 + 𝛾𝑖𝑗(d𝑥𝑖 +𝑁 𝑖 d𝑡

) (d𝑥𝑗 +𝑁 𝑗 d𝑡

), (7.1)

with the lapse 𝑁 , the shift 𝑁 𝑖 and the 3-dimensional space metric 𝛾𝑖𝑗 . In this section indices areraised and lowered with respect to 𝛾𝑖𝑗 and dots represent derivatives with respect to 𝑡. In termsof these variables, the action density for GR is

ℒGR =𝑀2

Pl

2

∫d𝑡(√−𝑔𝑅+ 𝜕𝑡

[√−𝑔[𝑘]

])(7.2)

=𝑀2

Pl

2

∫d𝑡𝑁√𝛾((3)𝑅[𝛾] + [𝑘]2 − [𝑘2]

), (7.3)

where (3)𝑅 is the three-dimensional scalar curvature built out of 𝛾 (no time derivatives in (3)𝑅)and 𝑘𝑖𝑗 is the three-dimensional extrinsic curvature,

𝑘𝑖𝑗 =1

2𝑁

(��𝑖𝑗 −∇(𝑖𝑁𝑗)

). (7.4)

The GR action can thus be expressed in a way which has no double or higher time derivatives andonly first time-derivatives squared of 𝛾𝑖𝑗 . This means that neither the shift nor the lapse are trulydynamical and they do not have any associated conjugate momenta. The conjugate momentumassociated with 𝛾 is,

𝑝𝑖𝑗 =𝜕√−𝑔𝑅𝜕��𝑖𝑗

. (7.5)

We can now construct the Hamiltonian density for GR in terms of the 12 phase space variables(𝛾𝑖𝑗 and 𝑝

𝑖𝑗 carry 6 component each),

ℋGR = 𝑁ℛ0(𝛾, 𝑝) +𝑁 𝑖ℛ𝑖(𝛾, 𝑝) . (7.6)

So we see that in GR, both the shift and the lapse play the role of Lagrange multipliers. Thusthey propagate a first-class constraint each which removes 2 phase space degrees of freedom perconstraint. The counting of the number of degrees of freedom in phase space thus goes as follows:

(2× 6)− 2 lapse constraints− 2× 3 shift constraints = 4 = 2× 2 , (7.7)

corresponding to a total of 4 degrees of freedom in phase space, or 2 independent degrees offreedom in field space. This is the very well-known and established result that in four dimensionsGR propagates 2 physical degrees of freedom, or gravitational waves have two polarizations.

This result is fully generalizable to any number of dimensions, and in 𝑑 spacetime dimensions,gravitational waves carry 𝑑(𝑑− 3)/2 polarizations. We now move to the case of massive gravity.



58 Claudia de Rham

7.1.2 ADM counting in massive gravity

We now amend the GR Lagrangian with a potential 𝒰 . As already explained, this can only beperformed by breaking covariance (with the exception of a cosmological constant). This potentialcould be a priori an arbitrary function of the metric, but contains no derivatives and so does notaffect the definition of the conjugate momenta 𝑝𝑖𝑗 This translates directly into a potential at thelevel of the Hamiltonian density,

ℋ = 𝑁ℛ0(𝛾, 𝑝) +𝑁 𝑖ℛ𝑖(𝛾, 𝑝) +𝑚2𝒰(𝛾𝑖𝑗 , 𝑁

𝑖, 𝑁), (7.8)

where the overall potential for ghost-free massive gravity is given in (6.4).If 𝒰 depends non-linearly on the shift or the lapse then these are no longer directly Lagrange

multipliers (if they are non-linear, they still appear at the level of the equations of motion, and sothey do not propagate a constraint for the metric but rather for themselves). As a result for anarbitrary potential one is left with (2× 6) degrees of freedom in the three-dimensional metric andits momentum conjugate and no constraint is present to reduce the phase space. This leads to 6degrees of freedom in field space: the two usual transverse polarizations for the graviton (as wehave in GR), in addition to two ‘vector’ polarizations and two ‘scalar’ polarizations.

These 6 polarizations correspond to the five healthy massive spin-2 field degrees of freedom inaddition to the sixth BD ghost, as explained in Section 2.5 (see also Section 7.2).

This counting is also generalizable to an arbitrary number of dimensions, in 𝑑 spacetime dimen-sions, a massive spin-2 field should propagate the same number of degrees of freedom as a masslessspin-2 field in 𝑑+ 1 dimensions, that is (𝑑+ 1)(𝑑− 2)/2 polarizations. However, an arbitrary po-tential would allow for 𝑑(𝑑−1)/2 independent degrees of freedom, which is 1 too many excitations,always corresponding to one BD ghost degree of freedom in an arbitrary number of dimensions.

The only way this counting can be wrong is if the constraints for the shift and the lapse cannotbe inverted for the shift and the lapse themselves, and thus at least one of the equations of motionfrom the shift or the lapse imposes a constraint on the three-dimensional metric 𝛾𝑖𝑗 . This loopholewas first presented in [138] and an example was provided in [137]. It was then used in [144] toexplain how the ‘no-go’ on the presence of a ghost in massive gravity could be circumvented.Finally, this argument was then carried through fully non-linearly in [295] (see also [342] for theanalysis in 1 + 1 dimensions as presented in [144]).

7.1.3 Eliminating the BD ghost

Linear Fierz–Pauli massive gravity

Fierz–Pauli massive gravity is special in that at the linear level (quadratic in the Hamiltonian),the lapse remains linear, so it still acts as a Lagrange multiplier generating a primary second-classconstraint. Defining the metric as ℎ𝜇𝜈 = 𝑀Pl(𝑔𝜇𝜈 − 𝜂𝜇𝜈), (where for simplicity and definitenesswe take Minkowski as the reference metric 𝑓𝜇𝜈 = 𝜂𝜇𝜈 , although most of what follows can be easilygeneralizable to an arbitrary reference metric 𝑓𝜇𝜈). Expanding the lapse as 𝑁 = 1 + 𝛿𝑁 , we haveℎ00 = 𝛿𝑁 + 𝛾𝑖𝑗𝑁

𝑖𝑁 𝑗 and ℎ0𝑖 = 𝛾𝑖𝑗𝑁𝑗 . In the ADM decomposition, the Fierz–Pauli mass term is

then (see Eq. (2.45))

𝒰 (2) = −𝑚−2ℒFPmass =1

8


)=

1

8

(ℎ2𝑖𝑗 − (ℎ𝑖𝑖)

2 − 2(𝑁2𝑖 − 𝛿𝑁ℎ𝑖𝑖

)), (7.9)

and is linear in the lapse. This is sufficient to deduce that it will keep imposing a constraint onthe three-dimensional phase space variables {𝛾𝑖𝑗 , 𝑝𝑖𝑗} and remove at least half of the unwanted BD



Massive Gravity 59

ghost. The shift, on the other hand, is non-linear already in the Fierz–Pauli theory, so their equa-tions of motion impose a relation for themselves rather than a constraint for the three-dimensionalmetric. As a result the Fierz–Pauli theory (at that order) propagates three additional degreesof freedom than GR, which are the usual five degrees of freedom of a massive spin-2 field. Non-linearly however the Fierz–Pauli mass term involve a non-linear term in the lapse in such a waythat the constraint associated with it disappears and Fierz–Pauli massive gravity has a ghost atthe non-linear level, as pointed out in [75]. This is in complete agreement with the discussion inSection 2.5, and is a complementary way to see the issue.

In Ref. [111], the most general potential was considered up to quartic order in the ℎ𝜇𝜈 , and itwas shown that there is no choice of such potential (apart from a pure cosmological constant) whichwould prevent the lapse from entering non-linearly. While this result is definitely correct, it doesnot however imply the absence of a constraint generated by the set of shift and lapse𝑁𝜇 = {𝑁,𝑁 𝑖}.Indeed there is no reason to believe that the lapse should necessarily be the quantity to generatesthe constraint necessary to remove the BD ghost. Rather it can be any combination of the lapseand the shift.

Example on how to evade the BD ghost non-linearly

As an instructive example presented in [137], consider the following Hamiltonian,

ℋ = 𝑁𝒞0(𝛾, 𝑝) +𝑁 𝑖𝒞𝑖(𝛾, 𝑝) +𝑚2𝒰 , (7.10)

with the following example for the potential

𝒰 = 𝑉 (𝛾, 𝑝)𝛾𝑖𝑗𝑁

𝑖𝑁 𝑗

2𝑁. (7.11)

In this example neither the lapse nor the shift enter linearly, and one might worry on the loss ofthe constraint to project out the BD ghost. However, upon solving for the shift and substitutingback into the Hamiltonian (this is possible since the lapse is not dynamical), we get

ℋ = 𝑁

(𝒞0(𝛾, 𝑝)−

𝛾𝑖𝑗𝒞𝑖𝒞𝑗2𝑚2𝑉 (𝛾, 𝑝)

), (7.12)

and the lapse now appears as a Lagrange multiplier generating a constraint, even though it was notlinear in (7.10). This could have been seen more easily, without the need to explicitly integratingout the shift by computing the Hessian

𝐿𝜇𝜈 =𝜕2ℋ

𝜕𝑁𝜇𝜕𝑁𝜈= 𝑚2 𝜕2𝒰

𝜕𝑁𝜇𝜕𝑁𝜈. (7.13)

In the example (7.10), one has

𝐿𝜇𝜈 =𝑚2𝑉 (𝛾, 𝑝)

𝑁3

(𝑁2𝑖 −𝑁 𝑁𝑖

−𝑁 𝑁𝑗 𝑁2 𝛾𝑖𝑗

)=⇒ det (𝐿𝜇𝜈) = 0 . (7.14)

The Hessian cannot be inverted, which means that the equations of motion cannot be solved forall the shift and the lapse. Instead, one of these ought to be solved for the three-dimensional phasespace variables which corresponds to the primary second-class constraint. Note that this constraintis not associated with a symmetry in this case and while the Hamiltonian is then pure constraintin this toy example, it will not be in general.

Finally, one could also have deduce the existence of a constraint by performing the linear changeof variable

𝑁𝑖 → 𝑛𝑖 =𝑁𝑖𝑁

, (7.15)



60 Claudia de Rham

in terms of which the Hamiltonian is then explicitly linear in the lapse,

ℋ = 𝑁

(𝒞0(𝛾, 𝑝) + 𝑛𝑖𝒞𝑖(𝛾, 𝑝) +𝑚2𝑉 (𝛾, 𝑝)

𝛾𝑖𝑗𝑛𝑖𝑛𝑗

2

), (7.16)

and generates a constraint that can be read for {𝑛𝑖, 𝛾𝑖𝑗 , 𝑝𝑖𝑗}.

Condition to evade the ghost

To summarize, the condition to eliminate (at least half of) the BD ghost is that the det of theHessian (7.13) 𝐿𝜇𝜈 vanishes as explained in [144]. This was shown to be the case in the ghost-freetheory of massive gravity (6.3) [(6.1)] exactly in some cases and up to quartic order, and then fullynon-linearly in [295]. We summarize the derivation in the general case in what follows.

Ultimately, this means that in massive gravity we should be able to find a new shift 𝑛𝑖 relatedto the original one as follows 𝑁 𝑖 = 𝑓0(𝛾, 𝑛) + 𝑁𝑓1(𝛾, 𝑛), such that the Hamiltonian takes thefollowing factorizable form

ℋ = (𝒜1(𝛾, 𝑝) +𝑁𝒞1(𝛾, 𝑝))ℱ(𝛾, 𝑝, 𝑛) + (𝒜2(𝛾, 𝑝) +𝑁𝒞2(𝛾, 𝑝)) . (7.17)

In this form, the equation of motion for the shift is manifestly independent of the lapse andintegrating over the shift 𝑛𝑖 manifestly keeps the Hamiltonian linear in the lapse and has theconstraint 𝒞1(𝛾, 𝑝)ℱ(𝛾, 𝑝, 𝑛𝑖(𝛾)) + 𝒞2(𝛾, 𝑝) = 0. However, such a field redefinition has not (yet)been found. Instead, the new shift 𝑛𝑖 found below does the next best thing (which is entirelysufficient) of a. Keeping the Hamiltonian linear in the lapse and b. Keeping its own equation ofmotion independent of the lapse, which is sufficient to infer the presence of a primary constraint.

Primary constraint

We now proceed by deriving the primary first-class constraint present in ghost-free (dRGT) massivegravity. The proof works equally well for any reference at no extra cost, and so we consider a generalreference metric 𝑓𝜇𝜈 in its own ADM decomposition, while keep the dynamical metric 𝑔𝜇𝜈 in itsoriginal ADM form (since we work in unitary gauge, we may not simplify the metric further),

𝑔𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −𝑁2 d𝑡2 + 𝛾𝑖𝑗(d𝑥𝑖 +𝑁 𝑖 d𝑡

) (d𝑥𝑗 +𝑁 𝑗 d𝑡

)(7.18)

𝑓𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −�� 2 d𝑡2 + 𝑓𝑖𝑗(d𝑥𝑖 + �� 𝑖 d𝑡

) (d𝑥𝑗 + �� 𝑗 d𝑡

), (7.19)

and denote again by 𝑝𝑖𝑗 the conjugate momentum associated with 𝛾𝑖𝑗 . 𝑓𝑖𝑗 is not dynamical inmassive gravity so there is no conjugate momenta associated with it. The bars on the referencemetric are there to denote that these quantities are parameters of the theory and not dynamicalvariables, although the proof for a dynamical reference metric and multi-gravity works equallywell, this is performed in Section 7.4.

Proceeding similarly as in the previous example, we perform a change of variables similar asin (7.15) (only more complicated, but which remains linear in the lapse when expressing 𝑁 𝑖 interms of 𝑛𝑖) [295, 296]

𝑁 𝑖 → 𝑛𝑖 defined as 𝑁 𝑖 − �� 𝑖 =(�� 𝛿𝑖𝑗 +𝑁𝐷𝑖

𝑗

)𝑛𝑗 , (7.20)

where the matrix 𝐷𝑖𝑗 satisfies the following relation

𝐷𝑖𝑘𝐷

𝑘𝑗 = (𝑃−1)𝑖𝑘𝛾

𝑘ℓ𝑓ℓ𝑗 , (7.21)

with𝑃 𝑖𝑗 = 𝛿𝑖𝑗 + (𝑛𝑖𝑓𝑗ℓ𝑛

ℓ − 𝑛𝑘𝑓𝑘ℓ𝑛ℓ𝛿𝑖𝑗) . (7.22)



Massive Gravity 61

In what follows we use the definition��𝑖

𝑗 = 𝜅𝐷𝑖𝑗 , (7.23)

with

𝜅 =√1− 𝑛𝑖𝑛𝑗𝑓𝑖𝑗 . (7.24)

The field redefinition naturally involves a square root through the expression of the matrix 𝐷in (7.21), which should come as no surprise from the square root structure of the potential term.For the potential to be writable in the metric language, the square root in the definition of thetensor 𝒦𝜇𝜈 should exist, which in turns imply that the square root in the definition of 𝐷𝑖

𝑗 in (7.21)must also exist. While complicated, the important point to notice is that this field redefinitionremains linear in the lapse (and so does not spoil the standard constraints of GR).

The Hamiltonian for massive gravity is then

ℋmGR = ℋGR +𝑚2𝒰= 𝑁 ℛ0(𝛾, 𝑝) +

(�� 𝑖 +

(�� 𝛿𝑖𝑗 +𝑁𝐷𝑖

𝑗

)𝑛𝑗)ℛ𝑖(𝛾, 𝑝) (7.25)

+𝑚2 𝒰(𝛾,𝑁 𝑖(𝑛), 𝑁) ,

where 𝒰 includes the new contributions from the mass term. 𝒰(𝛾,𝑁 𝑖, 𝑁) is neither linear in thelapse 𝑁 , nor in the shift 𝑁 𝑖. There is actually no choice of potential 𝒰 which would keep itlinear in the lapse beyond cubic order [111]. However, as we shall see, when expressed in terms ofthe redefined shift 𝑛𝑖, the non-linearities in the shift absorb all the original non-linearities in thelapse and 𝒰(𝛾, 𝑛𝑖, 𝑁). In itself this is not sufficient to prove the presence of a constraint, as theintegration over the shift 𝑛𝑖 could in turn lead to higher order lapse in the Hamiltonian,

𝒰(𝛾,𝑁 𝑖(𝑛𝑗), 𝑁) = 𝑁 𝒰0(𝛾, 𝑛𝑗) + �� 𝒰1(𝛾, 𝑛𝑗) , (7.26)

with

𝒰0 = −𝑀2Pl

4

√𝛾

3∑𝑛=0

(4− 𝑛)𝛽𝑛𝑛!

ℒ𝑛[��𝑖𝑗 ] (7.27)

𝒰1 = −𝑀2Pl

4

√𝛾(3!𝛽1𝜅+ 2𝛽2𝐷

𝑖𝑗𝑃

𝑗𝑖 (7.28)

+ 𝛽3𝜅[2𝐷

[𝑘𝑘𝑛

𝑖]𝑓𝑖𝑗𝐷𝑗ℓ𝑛ℓ +𝐷

[𝑖𝑖𝐷

𝑗]𝑗

] )− 𝑀2

Pl

4𝛽4

√𝑓 ,

where the 𝛽’s are expressed in terms of the 𝛼’s as in (6.28). For the purpose of this analysis it iseasier to work with that notation.

The structure of the potential is so that the equations of motion with respect to the shift areindependent of the lapse 𝑁 and impose the following relations in terms of ��𝑖 = 𝑛𝑗𝑓𝑖𝑗 ,

𝑚2√𝛾[3!𝛽1��𝑖 + 4𝛽2��

𝑗[𝑗 ��𝑖] + 𝛽3��

[𝑗𝑗

(��𝑘]𝑘��𝑖 − 2��

𝑘]𝑖 ��𝑘

)]= 𝜅ℛ𝑖(𝛾, 𝑝) , (7.29)

which entirely fixes the three shifts 𝑛𝑖 in terms of 𝛾𝑖𝑗 and 𝑝𝑖𝑗 as well as the reference metric 𝑓𝑖𝑗(note that �� 𝑖 entirely disappears from these equations of motion).

The two requirements defined previously are thus satisfied: a. The Hamiltonian is linear inthe lapse and b. the equations of motion with respect to the shift 𝑛𝑖 are independent of the lapse,which is sufficient to infer the presence of a primary constraint. This primary constraint is derivedby varying with respect to the lapse and evaluating the shift on the constraint surface (7.29),

𝒞0 = ℛ0(𝛾, 𝑝) +𝐷𝑖𝑗𝑛𝑗ℛ𝑖(𝛾, 𝑝) +𝑚2𝒰0(𝛾, 𝑛(𝛾, 𝑝)) ≈ 0 , (7.30)



62 Claudia de Rham

where the symbol “≈” means on the constraint surface. The existence of this primary constraintis sufficient to infer the absence of BD ghost. If we were dealing with a generic system (whichcould allow for some spontaneous parity violation), it could still be in principle that there are nosecondary constraints associated with 𝒞0 = 0 and the theory propagates 5.5 physical degrees offreedom (11 dofs in phase space). However, physically this never happens in the theory of gravitywe are dealing with preserves parity and is Lorentz invariant. Indeed, to have 5.5 physical degrees offreedom, one of the variables should have an equation of motion which is linear in time derivatives.Lorentz invariance then implies that it must also be linear in space derivatives which would thenviolate parity. However, this is only an intuitive argument and the real proof is presented below.Indeed, it ghost-free massive gravity admits a secondary constraint which was explicitly foundin [294].

Secondary constraint

Let us imagine we start with initial conditions that satisfy the constraints of the system, in partic-ular the modified Hamiltonian constraint (7.30). As the system evolves the constraint (7.30) needsto remain satisfied. This means that the modified Hamiltonian constraint ought to be independentof time, or in other words it should commute with the Hamiltonian. This requirement generates asecondary constraint,

𝒞2 ≡d

d𝑡𝒞0 = {𝒞0, 𝐻mGR} ≈ {𝒞0, 𝐻1} ≈ 0 , (7.31)

with 𝐻mGR,1 =∫d3𝑥ℋmGR,1 and

ℋ1 =(�� 𝑖 + ��𝑛𝑖(𝛾, 𝑝)

)ℛ𝑖 +𝑚2��𝒰1(𝛾, 𝑛(𝛾, 𝑝)) . (7.32)

Finding the precise form of this secondary constraint requires a very careful analysis of the Poissonbracket algebra of this system. This formidable task lead to some confusions at first (see Refs. [345])but was then successfully derived in [294] (see also [258, 259] and [343]). Deriving the whole setof Poisson brackets is beyond the scope of this review and we simply give the expression for thesecondary constraint,

𝒞2 ≡ 𝒞0∇𝑖(�� 𝑖 + ��𝑛𝑖

)+𝑚2��

(𝛾𝑖𝑗𝑝

ℓℓ − 2𝑝𝑖𝑗

)𝒰 𝑖𝑗1 (7.33)

+2𝑚2��√𝛾∇𝑖𝒰 𝑖1 𝑗𝐷𝑗𝑘𝑛

𝑘 +(ℛ𝑗𝐷𝑖

𝑘𝑛𝑘 −√𝛾ℬ 𝑖

𝑗

)∇𝑖(�� 𝑗 + ��𝑛𝑗

)+(∇𝑖ℛ0 +∇𝑖ℛ𝑗𝐷𝑗

𝑘𝑛𝑘) (�� 𝑖 + ��𝑛𝑖

),

where unless specified otherwise, all indices are raised and lowered with respect to the dynamicalmetric 𝛾𝑖𝑗 , and the covariant derivatives are also taken with respect to the same metric. We alsodefine

𝒰 𝑖𝑗1 =1√𝛾

𝜕𝒰1𝜕𝛾𝑖𝑗

(7.34)

ℬ𝑖𝑗 = −𝑀2

Pl

4

[(��−1)𝑘𝑗𝑓𝑖𝑘

(3𝛽1ℒ0[��] + 2𝛽2ℒ1[��] +

𝛽32ℒ2[��]

)(7.35)

−𝛽2𝑓𝑖𝑗 + 2𝛽3𝑓𝑖[𝑘��𝑘𝑗]

].

The important point to notice is that the secondary constraint (7.33) only depends on the phasespace variables 𝛾𝑖𝑗 , 𝑝

𝑖𝑗 and not on the lapse 𝑁 . Thus it constraints the phase space variables ratherthan the lapse and provides a genuine secondary constraint in addition to the primary one (7.30)(indeed one can check that 𝒞2|𝒞0=0 = 0.).



Massive Gravity 63

Finally, we should also check that this secondary constraint is also maintained in time. Thiswas performed [294], by inspecting the condition

d

d𝑡𝒞2 = {𝒞2, 𝐻mGR} ≈ 0 . (7.36)

This condition should be satisfied without further constraining the phase space variables, whichwould otherwise imply that fewer than five degrees of freedom are propagating. Since five fullyfledged dofs are propagating at the linearized level, the same must happen non-linearly.17 Ratherthan a constraint on {𝛾𝑖𝑗 , 𝑝𝑖𝑗}, (7.36) must be solved for the lapse. This is only possible if boththe two following conditions are satisfied

{𝒞2(𝑥),ℋ1(𝑦)} ≈/ 0 and {𝒞2(𝑥), 𝒞0(𝑦)} ≈/ 0 . (7.37)

As shown in [294], since these conditions do not vanish at the linear level (the constraints reduceto the Fierz–Pauli ones in that case), we can deduce that they cannot vanish non-linearly and thusthe condition (7.36) fixes the expression for the lapse rather than constraining further the phasespace dofs. Thus there is no tertiary constraint on the phase space.

To conclude, we have shown in this section that ghost-free (or dRGT) massive gravity is indeedfree from the BD ghost and the theory propagates five physical dofs about generic backgrounds.We now present the proof in other languages, but stress that the proof developed in this section issufficient to infer the absence of BD ghost.

Secondary constraints in bi- and multi-gravity

In bi- or multi-gravity where all the metrics are dynamical the Hamiltonian is pure constraint(every term is linear in the one of the lapses as can be seen explicitly already from (7.25) and(7.26)).

In this case, the evolution equation of the primary constraint can always be solved for theirrespective Lagrange multiplier (lapses) which can always be set to zero. Setting the lapses tozero would be unphysical in a theory of gravity and instead one should take a ‘bifurcation’ of theDirac constraint analysis as explained in [48]. Rather than solving for the Lagrange multipliers wecan choose to use the evolution equation of some of the primary constraints to provide additionalsecondary constraints instead of solving them for the lagrange multipliers.

Choosing this bifurcation leads to statements which are then continuous with the massivegravity case and one recovers the correct number of degrees of freedom. See Ref. [48] for anenlightening discussion.

7.2 Absence of ghost in the Stuckelberg language

7.2.1 Physical degrees of freedom

Another way to see the absence of ghost in massive gravity is to work directly in the Stuckelberglanguage for massive spin-2 fields introduced in Section 2.4. If the four scalar fields 𝜑𝑎 weredynamical, the theory would propagate six degrees of freedom (the two usual helicity-2 whichdynamics is encoded in the standard Einstein–Hilbert term, and the four Stuckelberg fields). Toremove the sixth mode, corresponding to the BD ghost, one needs to check that not all fourStuckelberg fields are dynamical but only three of them. See also [14] for a theory of two Stuckelbergfields.

17 Some dofs may ‘accidentally’ disappear about some special backgrounds, but dofs cannot disappear non-linearlyif they were present at the linearized level.



64 Claudia de Rham

Stated more precisely, in the Stuckelberg language beyond the DL, if ℰ𝑎 is the equation ofmotion with respect to the field 𝜑𝑎, the correct requirement for the absence of ghost is that theHessian 𝒜𝑎𝑏 defined as

𝒜𝑎𝑏 = −𝛿ℰ𝑎𝛿𝜑𝑏

=𝛿2ℒ

𝛿��𝑎𝛿��𝑏(7.38)

be not invertible, so that the dynamics of not all four Stuckelberg may be derived from it. This isthe case if

det (𝒜𝑎𝑏) = 0 , (7.39)

as first explained in Ref. [145]. This condition was successfully shown to arise in a number ofsituations for the ghost-free theory of massive gravity with potential given in (6.3) or equivalentlyin (6.1) in Ref. [145] and then more generically in Ref. [297].18 For illustrative purposes, we startby showing how this constraint arises in simple two-dimensional realization of ghost-free massivegravity before deriving the more general proof.

7.2.2 Two-dimensional case

Consider massive gravity on a two-dimensional space-time, d𝑠2 = −𝑁2 d𝑡2 + 𝛾 ( d𝑥+𝑁𝑥 d𝑡)2,

with the two Stuckelberg fields 𝜑0,1 [145]. In this case the graviton potential can only have oneindependent non-trivial term, (excluding the tadpole),

𝒰 = −𝑀2Pl

4𝑁√𝛾 (ℒ2(𝒦) + 1) . (7.40)

In light-cone coordinates,

𝜑± = 𝜑0 ± 𝜑1 (7.41)

𝒟± =1√𝛾𝜕𝑥 ±

1

𝑁

[𝜕𝑡 −𝑁𝑥𝜕𝑥

], (7.42)

the potential is thus

𝒰 = −𝑀2Pl

4𝑁√𝛾√(𝒟−𝜑−)(𝒟+𝜑+) . (7.43)

The Hessian of this Lagrangian with respect to the two Stuckelberg fields 𝜑± is then

𝒜𝑎𝑏 =𝛿2ℒmGR

𝛿��𝑎𝛿��𝑏= −𝑚2 𝛿2𝒰

𝛿��𝑎𝛿��𝑏

∝(

(𝒟−𝜑−)2 −(𝒟−𝜑

−)(𝒟+𝜑+)

−(𝒟−𝜑−)(𝒟+𝜑

+) (𝒟+𝜑+)2

), (7.44)

and is clearly non-invertible, which shows that not both Stuckelberg fields are dynamical. Inthis special case, the Hamiltonian is actually pure constraint as shown in [145], and there are nopropagating degrees of freedom. This is as expected for a massive spin-two field in two dimensions.

As shown in Refs. [144, 145] the square root can be traded for an auxiliary non-dynamicalvariable 𝜆𝜇𝜈 . In this two-dimensional example, the mass term (7.43) can be rewritten with the helpof an auxiliary non-dynamical variable 𝜆 as

𝒰 = −𝑀2Pl

4𝑁√𝛾

(𝜆+

1

2𝜆(𝒟−𝜑

−)(𝒟+𝜑+)

). (7.45)

A similar trick will be used in the full proof.

18 More recently, Alexandrov impressively performed the full analysis for bi-gravity and massive gravity in thevielbein language [15] determining the full set of primary and secondary constraints, confirming again the absenceof BD ghost. This resolves the potential sources of subtleties raised in Refs. [96, 351, 349, 348].



Massive Gravity 65

7.2.3 Full proof

The full proof in the minimal model (corresponding to 𝛼2 = 1 and 𝛼3 = −2/3 and 𝛼4 = 1/6in (6.3) or 𝛽2 = 𝛽3 = 0 in the alternative formulation (6.23)), was derived in Ref. [297]. We brieflyreview the essence of the argument, although the full technical derivation is beyond the scope ofthis review and refer the reader to Refs. [297] and [15] for a fully-fledged derivation.

Using a set of auxiliary variables 𝜆𝑎𝑏 (with 𝜆𝑎𝑏 = 𝜆𝑏𝑎, so these auxiliary variables contain tenelements in four dimensions) as explained previously, we can rewrite the potential term in theminimal model as [79, 342],

𝒰 =𝑀2

Pl

4

√−𝑔([𝜆] + [𝜆−1 · 𝑌 ]

), (7.46)

where the matrix 𝑌 has been defined in (2.77) and is equivalent to 𝑋 used previously. Upon inte-gration over the auxiliary variable 𝜆 we recover the square-root structure as mentioned in Ref. [144].We now perform an ADM decomposition as in (7.1) which implies the ADM decomposition on thematrix 𝑌 ,

𝑌 𝑎𝑏 = 𝑔𝜇𝜈𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑐𝑓𝑐𝑏 = −𝒟𝑡𝜑𝑎𝒟𝑡𝜑𝑐𝑓𝑐𝑏 + 𝑉 𝑎𝑏 , (7.47)

with

𝒟𝑡 =1

𝑁

(𝜕𝑡 −𝑁 𝑖𝜕𝑖

)(7.48)

𝑉 𝑎𝑏 = 𝛾𝑖𝑗𝜕𝑖𝜑𝑎𝜕𝑗𝜑

𝑐𝑓𝑐𝑏 . (7.49)

Since the matrix 𝑉 uses a projection along the 3 spatial directions it is genuinely a rank-3 matrixrather than rank 4. This implies that det𝑉 = 0. Notice that we consider an arbitrary referencemetric 𝑓 , as the proof does not depend on it and can be done for any 𝑓 at no extra cost [297]. Thecanonical momenta conjugate to 𝜑𝑎 is given by

𝑝𝑎 =1

2��(𝜆−1)𝑎𝑏𝒟0𝜑

𝑏 , (7.50)

with

�� = 2𝑀2Pl𝑚

2√𝛾 . (7.51)

In terms of these conjugate momenta, the equations of motion with respect to 𝜆𝑎𝑏 then imposesthe relation (after multiplying with the matrix19 𝛼𝜆 on both side),

𝜆𝑎𝑐𝐶𝑎𝑏𝜆𝑏𝑑 = 𝑉 𝑎𝑏 , (7.52)

with the matrix 𝐶𝑎𝑏 defined as

𝐶𝑎𝑏 = ��2𝑓𝑎𝑏 + 𝑝𝑎𝑝𝑏 . (7.53)

Since det𝑉 = 0, as mentioned previously, the equation of motion (7.52) is only consistent if wealso have det𝐶 = 0. This is the first constraint found in [297] which is already sufficient to remove(half) the BD ghost,

𝒞1 ≡det𝐶

det 𝑓= ��2 + (𝑓−1)𝑎𝑏𝑝𝑎𝑝𝑏 = 0 , (7.54)

19 We stress that multiplying with the matrix 𝜆 is not a projection, the equation (7.52) contains as much infor-mation as the equation of motion with respect to 𝜆, multiplying the with the matrix 𝜆 on both sides simply makethe rank of the equation more explicit.



66 Claudia de Rham

which is the primary constraint on a subset of physical phase space variables {𝛾𝑖𝑗 , 𝑝𝑎}, (by construc-tion det 𝑓 = 0). The secondary constraint is then derived by commuting 𝒞1 with the Hamiltonian.Following the derivation of [297], we get on the constraint surface

𝒞2 =1

��2𝑁

d𝒞1d𝑡

=1

��2𝑁

∫d𝑦 {𝒞1(𝑦), 𝐻(𝑥)} (7.55)

∝ −𝛾−1/2𝛾𝑖𝑗𝜋𝑖𝑗 − 2

��

𝛾(𝜆−1)𝑎𝑏𝜕𝑖𝜑

𝑎𝛾𝑖𝑗∇(𝑓)𝑗 𝑝𝑏 (7.56)

≡ 0 ,

where 𝜋𝑖𝑗 is the momentum conjugate associated with 𝛾𝑖𝑗 , and ∇(𝑓) is the covariant derivativeassociated with 𝑓 .

7.2.4 Stuckelberg method on arbitrary backgrounds

When working about different non-Minkowski backgrounds, one can instead generalize the defini-tion of the helicity-0 mode as was performed in [400]. The essence of the argument is to perform arotation in field space so that the fluctuations of the Stuckelberg fields about a curved backgroundform a vector field in the new basis, and one can then employ the standard treatment for a vectorfield. See also [10] for another study of the Stuckelberg fields in an FLRW background.

Recently, a covariant Stuckelberg analysis valid about any background was performed in Ref.[369] using the BRST formalism. Interestingly, this method also allows to derive the decouplinglimit of massive gravity about any background.

In what follows, we review the approach derived in [400] which provides yet another independentargument for the absence of ghost in all generalities. The proofs presented in Sections 7.1 and 7.2work to all orders about a trivial background while in [400], the proof is performed about a generic(curved) background, and the analysis can thus stop at quadratic order in the fluctuations. Bothtypes of analysis are equivalent so long as the fields are analytic, which is the case if one wishes toremain within the regime of validity of the theory.

Consider a generic background metric, which in unitary gauge (i.e., in the coordinate system {𝑥}where the Stuckelberg background fields are given by 𝜑𝑎(𝑥) = 𝑥𝜇𝛿𝑎𝜇), the background metric is given

by 𝑔bg𝜇𝜈 = 𝑒𝑎𝜇(𝑥)𝑒𝑏𝜈(𝑥)𝜂𝑎𝑏, and the background Stuckelberg fields are given by 𝜑𝑎bg(𝑥) = 𝑥𝑎−𝐴𝑎bg(𝑥).

We now add fluctuations about that background,

𝜑𝑎 = 𝜑𝑎bg − 𝑎𝑎 = 𝑥𝑎 −𝐴𝑎 (7.57)

𝑔𝜇𝜈 = 𝑔bg𝜇𝜈 + ℎ𝜇𝜈 , (7.58)

with 𝐴𝑎 = 𝐴𝑎bg + 𝑎𝑎.

Flat background metric

First, note that if we consider a flat background metric to start with, then at zeroth order in ℎ,the ghost-free potential is of the form [400], (this can also be seen from [238, 419])

ℒ𝐴 = −1

4𝐹𝐹 (1 + 𝜕𝐴+ · · · ) , (7.59)

with 𝐹𝑎𝑏 = 𝜕𝑎𝐴𝑏 − 𝜕𝑏𝐴𝑎. This means that for a symmetric Stuckelberg background configuration,i.e., if the matrix 𝜕𝜇𝜑

𝑎bg is symmetric, then 𝐹 bg

𝑎𝑏 = 0, and at quadratic order in the fluctuation 𝑎, theaction has a 𝑈(1)-symmetry. This symmetry is lost non-linearly, but is still relevant when lookingat quadratic fluctuations about arbitrary backgrounds. Now using the split about the background,



Massive Gravity 67

𝐴𝑎 = 𝐴𝑎bg + 𝑎𝑎, this means that up to quadratic order in the fluctuations 𝑎𝑎, the action at zerothorder in the metric fluctuation is of the form [400]

ℒ(2)𝑎 = ��𝜇𝛼𝜈𝛽𝑓𝜇𝜈𝑓𝛼𝛽 , (7.60)

with 𝑓𝜇𝜈 = 𝜕𝜇𝑎𝜈 − 𝜕𝜈𝑎𝜇 and ��𝜇𝛼𝜈𝛽 is a set of constant coefficients which depends on 𝐴𝑎bg. Thisquadratic action has an accidental 𝑈(1)-symmetry which is responsible for projecting out one ofthe four dofs naively present in the four Stuckelberg fluctuations 𝑎𝑎. Had we considered any otherpotential term, the 𝑈(1) symmetry would have been generically lost and all four Stuckelberg fieldswould have been dynamical.

Non-symmetric background Stuckelberg

If the background configuration is not symmetric, then at every point one needs to perform first aninternal Lorentz transformation Λ(𝑥) in the Stuckelberg field space, so as to align them with thecoordinate basis and recover a symmetric configuration for the background Stuckelberg fields. Inthis new Lorentz frame, the Stuckelberg fluctuation is ��𝜇 = Λ𝜇𝜈(𝑥)𝑎𝜈 . As a result, to quadratic orderin the Stuckelberg fluctuation the part of the ghost-free potential which is independent of the metricfluctuation and its curvature goes symbolically as (7.60) with 𝑓 replaced by 𝑓 → 𝑓 + (𝜕Λ)Λ−1��,(with 𝑓𝜇𝜈 = 𝜕𝜇��𝜈 − 𝜕𝜈 ��𝜇). Interestingly, the Lorentz boost (𝜕Λ)Λ−1 now plays the role of a massterm for what looks like a gauge field ��. This mass term breaks the 𝑈(1) symmetry, but thereis still no kinetic term for ��0, very much as in a Proca theory. This part of the potential is thusmanifestly ghost-free (in the sense that it provides a dynamics for only three of the four Stuckelbergfields, independently of the background).

Next, we consider the mixing with metric fluctuation ℎ while still assuming zero curvature. Atlinear order in ℎ, the ghost-free potential, (6.3) goes as follows

ℒ(2)𝐴ℎ = ℎ𝜇𝜈

3∑𝑛=1

𝑐𝑛𝑋(𝑛)𝜇𝜈 + ℎ𝐹 (𝜕𝐴+ · · · ) , (7.61)

where the tensors 𝑋(𝑛)𝜇𝜈 are similar to the ones found in the decoupling limit, but now expressed

in terms of the symmetric full four Stuckelberg fields rather than just 𝜋, i.e., replacing Π𝜇𝜈 by

𝜕𝜇𝐴𝜈 + 𝜕𝜈𝐴𝜇 in the respective expressions (8.29), (8.30) and (8.31) for 𝑋(1,2,3)𝜇𝜈 . Starting with the

symmetric configuration for the Stuckelberg fields, then since we are working at the quadratic level

in perturbations, one of the 𝐴𝜇 in the 𝑋(𝑛)𝜇𝜈 is taken to be the fluctuation 𝑎𝜇, while the others are

taken to be the background field 𝐴bg𝜇 . As a result in the first terms in ℎ𝑋 in (7.61) 𝜕0𝑎0 cannot

come at the same time as ℎ00 or ℎ0𝑖, and we can thus integrate by parts the time derivative actingon any 𝑎0, leading to a harmless first time derivative on ℎ𝑖𝑗 , and no time evolution for 𝑎0.

As for the second type of term in (7.61), since 𝐹 = 0 on the background field 𝐴bg𝜇 , the second

type of terms is forced to be proportional to 𝑓𝜇𝜈 and cannot involve any 𝜕0𝑎0 at all. As a result𝑎0 is not dynamical, which ensures that the theory is free from the BD ghost.

This part of the argument generalizes easily for non symmetric background Stuckelberg config-urations, and the same replacement 𝑓 → 𝑓 + (𝜕Λ)Λ−1�� still ensures that ��0 acquires no dynamicsfrom (7.61).

Background curvature

Finally, to complete the argument, we consider the effect from background curvature, then 𝑔bg𝜇𝜈 =𝜂𝜇𝜈 , with 𝑔

bg𝜇𝜈 = 𝑒𝑎𝜇(𝑥)𝑒

𝑏𝜈(𝑥). The space-time curvature is another source of ‘misalignment’ between

the coordinates and the Stuckelberg fields. To rectify for this misalignment, we could go two ways:



68 Claudia de Rham

Either perform a local change of coordinate so as to align the background metric 𝑔bg𝜇𝜈 with the flatreference metric 𝜂𝜇𝜈 (i.e., going to local inertial frame), or the other way around: i.e., express theflat reference metric in terms of the curved background metric, 𝜂𝑎𝑏 = 𝑒𝜇𝑎𝑒

𝜈𝑏𝑔

bg𝜇𝜈 , in terms of the

inverse vielbein, 𝑒𝜇𝑎 ≡ (𝑒−1)𝜇𝑎 . Then the building block of ghost-free massive gravity is the matrixX, defined previously as

X𝜇𝜈 =(𝑔−1𝜂

)𝜇𝜈= 𝑔𝜇𝛾(𝑒𝛼𝑎𝜕𝛾𝜑

𝑎)(𝑒𝛽𝑏𝜕𝜈𝜑𝑏)𝑔bg𝛼𝛽 . (7.62)

As a result, the whole formalism derived previously is directly applicable with the only subtletythat the Stuckelberg fields 𝜑𝑎 should be replaced by their ‘vielbein-dependent’ counterparts, i.e.,𝜕𝜇𝐴𝜈 → 𝑔bg𝜇𝜈 − 𝑔bg𝜈𝛼𝑒

𝛼𝑎𝜕𝜇𝜑

𝑎. In terms of the Stuckelberg field fluctuation 𝑎𝑎, this implies the

replacement 𝑎𝑎 → ��𝜇 = 𝑔bg𝜇𝜈𝑒𝜈𝑎𝑎𝑎, and symbolically, 𝑓 → 𝑓 + (𝜕Σ)Σ−1��, with Σ = 𝑔��. The

situation is thus the same as when we were dealing with a non-symmetric Stuckelberg backgroundconfiguration, after integration by parts (which might involve curvature harmless contributions),the potential can be written in a way which never involves any time derivative on ��0. As a result,��𝜇 plays the role of an effective Proca vector field which only propagates three degrees of freedom,and this about any curved background metric. The beauty of this argument lies in the correctidentification of the proper degrees of freedom when dealing with a curved background metric.

7.3 Absence of ghost in the vielbein formulation

Finally, we can also prove the absence of ghost for dRGT in the Vielbein formalism, either directlyat the level of the Lagrangian in some special cases as shown in [171] or in full generality in theHamiltonian formalism, as shown in [314]. The later proof also works in all generality for a multi-gravity theory and will thus be presented in more depth in what follows, but we first focus on aspecial case presented in Ref. [171].

Let us start with massive gravity in the vielbein formalism (6.1). As was the case in Part II,we work with the symmetric vielbein condition, 𝑒𝑎𝜇𝑓

𝑏𝜈𝜂𝑎𝑏 = 𝑒𝑎𝜈𝑓

𝑏𝜇𝜂𝑎𝑏. For simplicity we specialize

further to the case where 𝑓𝑎𝜇 = 𝛿𝑎𝜇, so that the symmetric vielbein condition imposes 𝑒𝑎𝜇 = 𝑒𝜇𝑎.Under this condition, the vielbein contains as many independent components as the metric. Thesymmetric veilbein condition ensures that one is able to reformulate the theory in a metric language.In 𝑑 spacetime dimensions, there is a priori 𝑑(𝑑+ 1)/2 independent components in the symmetricvielbein.

Varying the action (6.1) with respect to the vielbein leads to the modified Einstein equation,

𝐺𝑎 = 𝑡𝑎 = −𝑚2

2𝜀𝑎𝑏𝑐𝑑

(4𝑐0 𝑒

𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 + 3𝑐1 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑓𝑑 (7.63)

+2𝑐2 𝑒𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑 + 𝑐3 𝑓

𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑), (7.64)

with 𝐺𝑎 = 𝜀𝑎𝑏𝑐𝑑𝜔𝑏𝑐∧𝑒𝑑. From the Bianchi identity, 𝒟𝐺𝑎 = d𝐺𝑎−𝜔𝑏𝑎𝐺𝑏, we infer the 𝑑 constraints

𝒟𝑡𝑎 = d𝑡𝑎 − 𝜔𝑏𝑎𝑡𝑏 = 0 , (7.65)

leading to 𝑑(𝑑−1)/2 independent components in the vielbein. This is still one too many component,unless an additional constraint is found. The idea behind the proof in Ref. [171], is then to use theBianchi identities to infer an additional constraint of the form,

𝑚𝑎 ∧𝐺𝑎 = 𝑚𝑎 ∧ 𝑡𝑎 , (7.66)

where 𝑚𝑎 is an appropriate one-form which depends on the specific coefficients of the theory. Sucha constrain is present at the linear level for Fierz–Pauli massive gravity, and it was further shown



Massive Gravity 69

in Ref. [171] that special choices of coefficients for the theory lead to remarkably simple analogousrelations fully non-linearly. To give an example, we consider all the coefficients 𝑐𝑛 to vanish but𝑐1 = 0. In that case the Bianchi identity (7.65) implies

𝒟𝑡𝑎 = 0 =⇒ 𝜔𝑏𝑐𝑏 = 0 , (7.67)

where similarly as in (5.2), the torsionless connection is given in term of the vielbein as

𝜔𝑎𝑏𝜇 =1

2𝑒𝑐𝜇(𝑜

𝑎𝑏𝑐 − 𝑜 𝑎𝑏𝑐 − 𝑜𝑏 𝑎

𝑐 ) , (7.68)

with 𝑜𝑎𝑏𝑐 = 2𝑒𝑎𝜇𝑒𝑏𝜈 𝜕[𝜇𝑒𝜈]𝑐. The Bianchi identity (7.67) then implies 𝑒 𝑏𝑎 𝜕[𝑏𝑒𝑎𝑎] = 0, so that we

obtain an extra constraint of the form (7.66) with 𝑚𝑎 = 𝑒𝑎. Ref. [171] derived similar constraintsfor other parameters of the theory.

7.4 Absence of ghosts in multi-gravity

We now turn to the proof for the absence of ghost in multi-gravity and follow the vielbein formu-lation of Ref. [314]. In this subsection we use the notation that uppercase Latin indices represent𝑑-dimensional Lorentz indices, 𝐴,𝐵, · · · = 0, · · · , 𝑑− 1, while lowercase Latin indices represent the𝑑− 1-dimensional Lorentz indices along the space directions after ADM decomposition, 𝑎, 𝑏, · · · =1, · · · , 𝑑−1. Greek indices represent 𝑑-dimensional spacetime indices 𝜇, 𝜈 = 0, · · · , 𝑑−1, while the‘middle’ of the Latin alphabet indices 𝑖, 𝑗 · · · represent pure space indices 𝑖, 𝑗 · · · = 1, · · · , 𝑑 − 1.Finally, capital indices label the metric and span over 𝐼, 𝐽,𝐾, · · · = 1, · · · , 𝑁 .

Let us start with 𝑁 non-interacting spin-2 fields. The theory has then 𝑁 copies of coordinatetransformation invariance (the coordinate system associated with each metric can be changedseparately), as well as 𝑁 copies of Lorentz invariance. At this level may, for each vielbein 𝑒(𝐽),𝐽 = 1, · · · , 𝑁 we may use part of the Lorentz freedom to work in the upper triangular form forthe vielbein,

𝑒(𝐽)𝐴𝜇 =

(𝑁(𝐽) 𝑁

𝑖(𝐽)𝑒(𝐽)

𝑎𝑖

0 𝑒(𝐽)𝑎𝑖

), 𝑒(𝐽)

𝜇𝐴 =

(𝑁(𝐽)

−1 0−𝑁 𝑖

(𝐽)𝑁−1 𝑒(𝐽)

𝑖𝑎

), (7.69)

leading to the standard ADM decomposition for the metric,

𝑔(𝐽)𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = 𝑒(𝐽)𝐴𝜇𝑒(𝐽)

𝐵𝜈𝜂𝐴𝐵 d𝑥𝜇 d𝑥𝜈

= −𝑁(𝐽)2 d𝑡2 + 𝛾(𝐽)𝑖𝑗

(d𝑥𝑖 +𝑁 𝑖

(𝐽) d𝑡) (

d𝑥𝑗 +𝑁(𝐽)𝑗 d𝑡

), (7.70)

with the three-dimensional metric 𝛾(𝐽)𝑖𝑗 = 𝑒(𝐽)𝑎𝑖𝑒(𝐽)

𝑏𝑗𝛿𝑎𝑏. Starting with non-interacting fields, we

simply take 𝑁 copies of the GR action,

𝐿𝑁GR =

∫d𝑡

𝑁∑𝐽=1

√−𝑔(𝐽)𝑅(𝐽) , (7.71)

and the Hamiltonian in terms of the vielbein variables then takes the form (7.6)

ℋ𝑁GR =

∫d𝑑𝑥

𝑁∑𝐽=1

(𝜋(𝐽)

𝑖𝑎 ��(𝐽)

𝑎𝑖 +𝑁(𝐽)𝒞(𝐽)0 +𝑁 𝑖

(𝐽)𝒞(𝐽)𝑖 −1

2𝜆𝑎𝑏(𝐽)𝒫(𝐽)𝑎𝑏

), (7.72)

where 𝜋(𝐽)𝑖𝑎 is the conjugate momentum associated with the vielbein 𝑒(𝐽)

𝑎𝑖 and the constraints

𝒞(𝐽)0,𝑖 = 𝒞0,𝑖(𝑒(𝐽), 𝜋(𝐽)) are the ones mentioned previously in (7.6) (now expressed in the vielbein



70 Claudia de Rham

variables) and are related to diffeomorphism invariance. In the vielbein language there is anaddition 𝑑(𝑑− 1)/2 primary constraints for each vielbein field

𝒫(𝐽)𝑎𝑏 = 𝑒(𝐽)[𝑎𝑖 𝜋(𝐽)𝑖𝑏] , (7.73)

related to the residual local Lorentz symmetry still present after fixing the upper triangular formfor the vielbeins.

Now rather than setting part of the 𝑁 Lorentz frames to be on the upper diagonal form for allthe 𝑁 vielbein (7.69) we only use one Lorentz boost to set one of the vielbein in that form, say𝑒(1), and ‘unboost’ the 𝑁 − 1 other frames, so that for any of the other vielbein one has

𝑒(𝐽)𝐴𝜇 =

(𝑁(𝐽)𝛾(𝐽) +𝑁 𝑖

(𝐽)𝑒(𝐽)𝑎𝑖𝑝(𝐽)𝑎 𝑁(𝐽)𝑝

𝑎(𝐽) +𝑁 𝑖

(𝐽)𝑒(𝐽)𝑏𝑖𝑆(𝐽)

𝑎𝑏

𝑒(𝐽)𝑎𝑖 𝑒(𝐽)

𝑏𝑖𝑆(𝐽)

𝑎𝑏

)(7.74)

𝑆(𝐽)𝑎𝑏 = 𝛿𝑎𝑏 + 𝛾−1

(𝐽)𝑝𝑎(𝐽)𝑝(𝐽)𝑏 (7.75)

𝛾(𝐽) =√

1 + 𝑝(𝐽)𝑎𝑝𝑎(𝐽) (7.76)

where 𝑝(𝐽)𝑎 is the boost that would bring that vielbein in the upper diagonal form.We now consider arbitrary interactions between the 𝑁 fields of the form (6.1),

𝐿𝑁 int =

𝑁∑𝐽1,··· ,𝐽𝑑=1

𝛼𝐽1,··· ,𝐽𝑑𝜀𝑎1···𝑎𝑑 𝑒𝑎1(𝐽1)∧ · · · ∧ 𝑒𝑎𝑑(𝐽𝑑) , (7.77)

where for concreteness we assume 𝑑 ≤ 𝑁 , otherwise the formalism is exactly the same (there issome redundancy in this formulation, i.e., some interactions are repeated in this formulation, butthis has no consequence for the argument). Since the vielbeins 𝑒(𝐽)

𝐴0 are linear in their respective

shifts and lapse 𝑁(𝐽), 𝑁𝑖(𝐽) and the vielbeins 𝑒(𝐽)

𝐴𝑖 do not depend any shift nor lapse, it is easy to

see that the general set of interactions (7.77) lead to a Hamiltonian which is also linear in everyshift and lapse,

ℋ𝑁 int =𝑁∑𝐽=1

(𝑁(𝐽)𝒞int(𝐽)(𝑒, 𝑝) +𝑁 𝑖

(𝐽)𝒞int(𝐽)𝑖(𝑒, 𝑝)

). (7.78)

Indeed the wedge structure of (6.1) or (7.77) ensures that there is one and only one vielbein withtime-like index 𝑒(𝐽)

𝐴0 for every term 𝜀𝑎1···𝑎𝑑 𝑒

𝑎1(𝐽1)∧ · · · ∧ 𝑒𝑎𝑑(𝐽𝑑).

Notice that for the interactions, the terms 𝒞int(𝐽)0,𝑖 can depend on all the 𝑁 vielbeins 𝑒(𝐽′) and

all the 𝑁 − 1 ‘boosts’ 𝑝(𝐽′), (as mentioned previously, part of one Lorentz frame is set so that𝑝(1) = 0 and 𝑒(1) is in the upper diagonal form). Following the procedure of [314], we can nowsolve for the 𝑁 − 1 remaining boosts by using (𝑁 − 1) of the 𝑁 shift equations of motion

𝒞(𝐽) 𝑖(𝑒, 𝜋) + 𝒞(𝐽) 𝑖(𝑒, 𝑝) = 0 ∀ 𝐽 = 1, · · · , 𝑁 . (7.79)

Now assuming that all 𝑁 vielbein are interacting,20 (i.e., there is no vielbein 𝑒(𝐽) which doesnot appear at least once in the interactions (7.77) which mix different vielbeins), the shift equa-tions (7.79) will involve all the 𝑁 − 1 boosts and can be solved for them without spoiling thelinearity in any of the 𝑁 lapses 𝑁(𝐽). As a result, the 𝑁 − 1 lapses 𝑁(𝐽) for 𝐽 = 2, · · · , 𝑁 areLagrange multiplier for (𝑁−1) first class constraints. The lapse 𝑁(1) for the first vielbein combineswith the remaining shift 𝑁 𝑖

(1) to generate the one remaining copy of diffeomorphism invariance.

20 If only 𝑀 vielbein of the 𝑁 vielbein are interacting there will be (𝑁−𝑀+1) copies of diffeomorphism invarianceand 𝑀−1 additional Hamiltonian constraints, leading to the correct number of dofs for (𝑁−𝑀+1) massless spin-2fields and 𝑀 − 1 massive spin-2 fields.



Massive Gravity 71

We now have all the ingredients to count the number of dofs in phase space: We start with𝑑2 components in each of the 𝑁 vielbein 𝑒 𝑎(𝐽) 𝑖 and associated conjugate momenta, that is a total

of 2 × 𝑑2 × 𝑁 phase space variables. We then have 2 × 𝑑(𝑑 − 1)/2 × 𝑁 constraints21 associatedwith the 𝜆𝑎𝑏(𝐽). There is one copy of diffeomorphism removing 2 × (𝑑 + 1) phase space dofs (with

Lagrange multiplier 𝑁(1) and 𝑁 𝑖(1)) and (𝑁 − 1) additional first-class constraints with Lagrange

multipliers 𝑁(𝐽≥2) removing 2× (𝑁 − 1) dofs. As a result we end up with(2× 𝑑(𝑑− 1)

2×𝑁

)− 2× 𝑑(𝑑− 1)

2×𝑁 − 2× (𝑑+ 1)− 2× (𝑁 − 1)

=(𝑑2𝑁 − 2𝑁 + 𝑑(𝑁 − 2)

)phase space dofs

=1

2

(𝑑2𝑁 − 2𝑁 + 𝑑(𝑁 − 2)

)field space dofs (7.80)

=1

2

(𝑑2 − 𝑑− 2

)dofs for a massless spin-2 field

+1

2

(𝑑2 + 𝑑− 2

)× (𝑁 − 1) dofs for (𝑁 − 1) a massive spin-2 fields , (7.81)

which is the correct counting in (𝑑 + 1) spacetime dimensions, and the theory is thus free of anyBD ghost.

8 Decoupling Limits

8.1 Scaling versus decoupling

Before moving to the decoupling of massive gravity and bi-gravity, let us make a brief interludeconcerning the correct identification of degrees of freedom. The Stuckelberg trick used previouslyto identify the correct degrees of freedom works in all generality, but care must be used whentaking a “decoupling limit” (i.e., scaling limit) as will be done in Section 8.2.

Imagine the following gauge field theory

ℒ = −1

2𝑚2𝐴𝜇𝐴

𝜇 , (8.1)

i.e., the Proca mass term without any kinetic Maxwell term for the gauge field. Since there are nodynamics in this theory, there is no degrees of freedom. Nevertheless, one could still proceed anduse the same split 𝐴𝜇 = 𝐴⊥

𝜇 + 𝜕𝜇𝜒/𝑚 as performed previously,

ℒ = −1

2𝑚2𝐴⊥

𝜇𝐴⊥𝜇 +𝑚(𝜕𝜇𝐴

⊥𝜇)𝜒− 1

2(𝜕𝜒)2 , (8.2)

so as to introduce what appears to be a kinetic term for the mode 𝜒. At this level the theory is stillinvariant under 𝜒→ 𝜒+𝑚𝜉 and 𝐴⊥

𝜇 → 𝐴⊥𝜇 − 𝜕𝜇𝜉, and so while there appears to be a dynamical

degree of freedom 𝜒, the symmetry makes that degree of freedom unphysical, so that (8.2) stillpropagates no physical degree of freedom.

Now consider the 𝑚 → 0 scaling limit of (8.2) while keeping 𝐴⊥𝜇 and 𝜒 finite. In that scaling

limit, the theory reduces to

ℒ𝑚→0 = −1

2(𝜕𝜒)2 , (8.3)

21 Technically, only one of them generates a first class constraint, while the 𝑁 − 1 others generate a second-classconstraint. There are, therefore, (𝑁 − 1) additional secondary constraints to be found by commuting the primaryconstraint with the Hamiltonian, but the presence of these constraints at the linear level ensures that they mustexist at the non-linear level. There is also another subtlety in obtaining the secondary constraints associated withthe fact that the Hamiltonian is pure constraint, see the discussion in Section 7.1.3 for more details.



72 Claudia de Rham

i.e., one degree of freedom with no symmetry which implies that the theory (8.3) propagates onedegree of freedom. This is correct and thus means that (8.3) is not a consistent decoupling limitof (8.2) since the number of degrees of freedom is different already at the linear level. In the rest ofthis review, we will call a decoupling limit a specific type of scaling limit which preserves the samenumber of physical propagating degrees of freedom in the linear theory. As suggested by the name,a decoupling limit is a special kind of limit in which some of the degrees of freedom of the originaltheory might decouple from the rest, but the total number of degrees of freedom remains identical.For the theory (8.2), this means that the scaling ought to be taken not with 𝐴⊥

𝜇 fixed but rather

with 𝐴⊥𝜇 = 𝐴⊥

𝜇 /𝑚 fixed. This is indeed a consistent rescaling which leads to finite contributionsin the limit 𝑚→ 0,

ℒ𝑚→0 = −1

2𝐴⊥𝜇𝐴

⊥𝜇 + (𝜕𝜇𝐴⊥𝜇)𝜒− 1

2(𝜕𝜒)2 , (8.4)

which clearly propagates no degrees of freedom.This procedure is true in all generality: a decoupling limit is a special scaling limit where all

the fields in the original theory are scaled with the highest possible power of the scale in such away that the decoupling limit is finite.

A decoupling limit of a theory never changes the number of physical degrees of freedom of atheory. At best it ‘decouples’ some of them in such a way that they are inaccessible from anothersector.

Before looking at the massive gravity limit of bi-gravity and other decoupling limits of massiveand bi-gravity, let us start by describing the different scaling limits that can be taken. We startwith a bi-gravity theory where the two spin-2 fields have respective Planck scales 𝑀𝑔 and 𝑀𝑓 andthe interactions between the two metrics arises at the scale 𝑚. In order to stick to the relevantpoints we perform the analysis in four dimensions, but the following arguments extend trivially toarbitrary dimensions.

∙ Non-interacting Limit: The most natural question to ask is what happens in the limitwhere the interactions between the two fields are ‘switched off’, i.e., when sending the scale𝑚→ 0, (the limit 𝑚→ 0 is studied more carefully in Sections 8.3 and 8.4). In that case if thetwo Planck scales 𝑀𝑔,𝑓 remain fixed as 𝑚→ 0, we then recover two massless non-interactingspin-2 fields (carrying both 2 helicity-2 modes), in addition to a decoupled sector containinga helicity-0 mode and a helicity-1 mode. In bi-gravity matter fields couple only to one metric,and this remains the case in the limit 𝑚 → 0, so that the two massless spin-2 fields live intwo fully decoupled sectors even when matter in included.

∙ Massive Gravity: Alternatively, we may look at the limit where one of the spin-2 fields(say 𝑓𝜇𝜈) decouples. This can be studied by sending its respective Planck scale to infinity.The resulting limit corresponds to a massive spin-2 field (carrying five dofs) and a decoupledmassless spin-2 field carrying 2 dofs. This is nothing other than the massive gravity limit ofbi-gravity (which includes a fully decoupled massless sector).

If one considers matter coupling to the metric 𝑓𝜇𝜈 which scales in such a way that a non-trivial solution for 𝑓𝜇𝜈 survives in the 𝑀𝑓 → ∞ limit 𝑓𝜇𝜈 → 𝑓𝜇𝜈 , we then obtain a massivegravity sector on an arbitrary non-dynamical reference metric 𝑓𝜇𝜈 . The dynamics of themassless spin-2 field fully decouples from that of the massive sector.

∙ Other Decoupling Limits Finally, one can look at combinations of the previous limits,and the resulting theory depends on how fast 𝑀𝑓 ,𝑀𝑔 → ∞ compared to how fast 𝑚 → 0.For instance if one takes the limit 𝑀𝑓 ,𝑀𝑔 → ∞ and 𝑚 → 0, while keeping both 𝑀𝑔/𝑀𝑓

and Λ33 = 𝑀𝑔𝑚

2 fixed, then we obtain what is called the Λ3-decoupling limit of bi-gravity(derived in Section 8.4), where the dynamics of the two helicity-2 modes (which are both



Massive Gravity 73

massless in that limit), and that of the helicity-1 and -0 modes can be followed withoutkeeping track of the standard non-linearities of GR.

If on top of this Λ3-decoupling limit one further takes 𝑀𝑓 → ∞, then one of the masslessspin-2 fields fully decoupled (no communication between that field and the helicity-1 and -0modes). If, on the other hand, we take the additional limit𝑚→ 0 on top of the Λ3-decouplinglimit, then the helicity-0 and -1 modes fully decouple from both helicity-2 modes.

In all of these decoupling limits, the number of dofs remains the same as in the original theory,some fields are simply decoupled from the rest of the standard gravitational sector. These preventsany communication between these decoupled fields and the gravitational sector, and so from thegravitational sector view point it appears as if these decoupled fields did not exist.

It is worth stressing that all of these limits are perfectly sensible and lead to sensible theories,(from a theoretical view point). This is important since if one of these scaling limits lead to apathological theory, it would have severe consequences for the parent bi-gravity theory itself.

Similar decoupling limit could be taken in multi-gravity and out of 𝑁 interacting spin-2 fields,we could obtain for instance 𝑁 decoupled massless spin-2 fields and 3(𝑁 − 1) decoupled dofs inthe helicity-0 and -1 modes.

In what follows we focus on massive gravity limit of bi-gravity when 𝑀𝑓 →∞.

8.2 Massive gravity as a decoupling limit of bi-gravity

8.2.1 Minkowski reference metric

In the following two sections we review the decoupling arguments given previously in the literature,(see for instance [154]). We start with the theory of bi-gravity presented in Section 5.4 with theaction (5.43)

ℒbi−gravity =𝑀2𝑔

2

√−𝑔𝑅[𝑔] +

𝑀2𝑓

2

√−𝑓𝑅[𝑓 ] + 1

4𝑚2𝑀2

Pl

√−𝑔ℒ𝑚(𝑔, 𝑓)

+√−𝑔ℒ(matter)

𝑔 (𝑔𝜇𝜈 , 𝜓𝑔) +√−𝑓ℒ(matter)

𝑓 (𝑓𝜇𝜈 , 𝜓𝑓 ) , (8.5)

with ℒ𝑚(𝑔, 𝑓) =∑4𝑛=0 𝛼𝑛ℒ𝑛[𝒦(𝑔, 𝑓)] as defined in (6.3) and where 𝒦𝜇𝜈 = 𝛿𝜇𝜈 −

√𝑔𝜇𝛼𝑓𝛼𝜈 . We also

allow for the coupling to matter with different species 𝜓𝑔,𝑓 living on each metrics.We now consider matter fields 𝜓𝑓 such that 𝑓𝜇𝜈 = 𝜂𝜇𝜈 is a solution to the equations of motion

(so for instance there is no overall cosmological constant living on the metric 𝑓𝜇𝜈). In that case wecan write that metric 𝑓𝜇𝜈 as


𝑀𝑓𝜒𝜇𝜈 , (8.6)

We may now take the limit𝑀𝑓 →∞, while keeping the scales𝑀𝑔 and𝑚 and all the fields 𝜒, 𝑔, 𝜓𝑓,𝑔fixed. We then recover massive gravity plus a completely decoupled massless spin-2 field 𝜒𝜇𝜈 , anda fully decoupled matter sector 𝜓𝑓 living on flat space

ℒbi−gravity𝑀𝑓→∞−−−−−→ ℒMG(𝑔, 𝜂) +

√−𝑔ℒ(matter)

𝑔 (𝑔𝜇𝜈 , 𝜓𝑔) (8.7)

+1

2𝜒𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝜒𝛼𝛽 + ℒ(matter)

𝑓 (𝜂𝜇𝜈 , 𝜓𝑓 ) ,

with the massive gravity Lagrangian ℒMG is expressed in (6.3). That massive gravity Lagrangianremains fully non-linear in this limit and is expressed in terms of the full metric 𝑔𝜇𝜈 and thereference metric 𝜂𝜇𝜈 . While the metric 𝑓𝜇𝜈 is ‘frozen’ in this limit, we emphasize however thatthe massless spin-2 field 𝜒𝜇𝜈 is itself not frozen – its dynamics is captured through the kinetic



74 Claudia de Rham

term 𝜒𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝜒𝛼𝛽 , but that spin-2 field decouple from its own matter sector 𝜓𝑓 , (although this canbe accommodated for by scaling the matter fields 𝜓𝑓 accordingly in the limit 𝑀𝑓 → ∞ so as tomaintain some interactions).

At the level of the equations of motion, in the limit 𝑀𝑓 → ∞ we obtain the massive gravitymodified Einstein equation for 𝑔𝜇𝜈 , the free massless linearized Einstein equation for 𝜒𝜇𝜈 whichfully decouples and the equation of motion for all the matter fields 𝜓𝑓 on flat spacetime, (see alsoRef. [44]).

8.2.2 (A)dS reference metric

To consider massive gravity with an (A)dS reference metric as a limit of bi-gravity, we include acosmological constant for the metric 𝑓 into (8.5)

ℒCC,f = −𝑀2𝑓

∫d4𝑥√−𝑓Λ𝑓 . (8.8)

There can also be in principle another cosmological constant living on top of the metric 𝑔𝜇𝜈 butthis can be included into the potential 𝒰(𝑔, 𝑓). The background field equations of motion are thengiven by

𝑀2𝑓𝐺𝜇𝜈 [𝑓 ] +

𝑚2𝑀2Pl

4√−𝑓

(𝛿

𝛿𝑓𝜇𝜈√−𝑔 𝒰(𝑔, 𝑓)

)= 𝑇𝜇𝜈(𝜓𝑓 )−𝑀2

𝑓Λ𝑓𝑓𝜇𝜈 (8.9)

𝑀2Pl𝐺𝜇𝜈 [𝑔] +

𝑚2𝑀2Pl

4√−𝑔

(𝛿

𝛿𝑔𝜇𝜈√−𝑔 𝒰(𝑔, 𝑓)

)= 𝑇𝜇𝜈(𝜓𝑔) . (8.10)

Taking now the limit 𝑀𝑓 →∞ while keeping the cosmological constant Λ𝑓 fixed, the backgroundsolution for the metric 𝑓𝜇𝜈 is nothing other than dS (or AdS depending on the sign of Λ𝑓 ). So wecan now express the metric 𝑓𝜇𝜈 as

𝑓𝜇𝜈 = 𝛾𝜇𝜈 +1

𝑀𝑓𝜒𝜇𝜈 , (8.11)

where 𝛾𝜇𝜈 is the dS metric with Hubble parameter 𝐻 =√Λ𝑓/3. Taking the limit 𝑀𝑓 → ∞, we

recover massive gravity on (A)dS plus a completely decoupled massless spin-2 field 𝜒𝜇𝜈 ,

ℒbi−gravity −𝑀2𝑓

∫d4𝑥√−𝑓Λ𝑓

𝑀𝑓→∞−−−−−→ 𝑀2Pl

2

√−𝑔𝑅+

𝑚2

4𝒰(𝑔, 𝛾) (8.12)

+1

2𝜒𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝜒𝛼𝛽 ,

where once again the scales 𝑀Pl and 𝑚 are kept fixed in the limit 𝑀𝑓 → ∞. 𝛾𝜇𝜈 now plays therole of a non-trivial reference metric for massive gravity. This corresponds to a theory of massivegravity on a more general reference metric as presented in [296]. Here again the Lagrangian formassive gravity is given in (6.3) with now 𝒦𝜇𝜈 (𝑔) = 𝛿𝜇𝜈 −

√𝑔𝜇𝛼𝛾𝛼𝜈 . The massive gravity action

remains fully non-linear in the limit𝑀𝑓 →∞ and is expressed solely in terms of the full metric 𝑔𝜇𝜈and the reference metric 𝛾𝜇𝜈 , while the excitations 𝜒𝜇𝜈 for the massless graviton remain dynamicalbut fully decouple from the massive sector.

8.2.3 Arbitrary reference metric

As is already clear from the previous discussion, to recover massive gravity on a non-trivial referencemetric as a limit of bi-gravity, one needs to scale the Matter Lagrangian that couples to what will



Massive Gravity 75

become the reference metric (say the metric 𝑓 for definiteness) in such a way that the Riemanncurvature of 𝑓 remains finite in that decoupling limit. For a macroscopic description of the matterliving on 𝑓 this is in principle always possible. For instance one can consider a point source ofmass 𝑀BH living on the metric 𝑓 . Then, taking the limit 𝑀𝑓 ,𝑀BH → ∞ while keeping the ratio𝑀BH/𝑀𝑓 fixed, leads to a theory of massive gravity on a Schwarzschild reference metric and adecoupled massless graviton. However, some care needs to be taken to see how this works whenthe dynamics of the matter sourcing 𝑓 is included.

As soon as the dynamics of the matter field is considered, one has to send the scale of that fieldto infinity so that it maintains some nonzero effect on 𝑓 in the limit 𝑀𝑓 →∞, i.e.,

lim𝑀𝑓→∞

1

𝑀2𝑓

𝑇𝜇𝜈 = lim𝑀𝑓→∞

1√−𝑓𝑀2

𝑓

𝛿√−𝑓ℒ(matter)

𝑓

𝛿𝑓𝜇𝜈→ finite . (8.13)

Nevertheless, this can be achieved in such a way that the fluctuations of the matter fields remainfinite and decouple in the limit 𝑀𝑓 →∞. We note that this scaling is the key difference betweenthe decoupling limit of bi-gravity on a Minkowski reference metric derived in section 8.2.1 where thematter field scale as lim𝑀𝑓→∞

1𝑀2

𝑓𝑇𝜇𝜈 → 0 and the decoupling limit of bi-gravity on an arbitrary

reference metric derived here.As an example, suppose that the Lagrangian for the matter (for example a scalar field) sourcing

the 𝑓 metric is

ℒ(matter)𝑓 =

√−𝑓(−1

2𝑓𝜇𝜈𝜕𝜇𝜒𝜕𝜈𝜒− 𝑉0𝐹

(𝜒𝜆

))(8.14)

where 𝐹 (𝑋) is an arbitrary dimensionless function of its argument. Then choosing 𝜒 to take theform

𝜒 =𝑀𝑓 ��+ 𝛿𝜒 , (8.15)

and rescaling 𝑉0 = 𝑀2𝑓𝑉0 and 𝜆 = 𝑀𝑓 ��, then on taking the limit 𝑀𝑓 → ∞ keeping ��, 𝛿𝜒, �� and

𝑉0 fixed, since

ℒ(matter)𝑓 →𝑀2

𝑓

√−𝑓(−1

2𝑓𝜇𝜈𝜕𝜇��𝜕𝜈 ��− 𝑉0𝐹

( ��

))+ fluctuations , (8.16)

we find that the background stress energy blows up in such a way that 1𝑀2

𝑓𝑇𝜇𝜈 remains finite

and nontrivial, and in addition the background equations of motion for �� remain well-defined andnontrivial in this limit,

2𝑓 �� =𝑉0��𝐹 ′( ��

). (8.17)

This implies that even in the limit 𝑀𝑓 → ∞, 𝑓𝜇𝜈 can remain consistently as a nontrivial sourcedmetric which is a solution of some dynamical equations sourced by matter. In addition the actionfor the fluctuations 𝛿𝜒 asymptotes to a free theory which is coupled only to the fluctuations of 𝑓𝜇𝜈which are themselves completely decoupled from the fluctuations of the metric 𝑔 and matter fieldscoupled to 𝑔.

As a result, massive gravity with an arbitrary reference metric can be seen as a consistentlimit of bi-gravity in which the additional degrees of freedom in the 𝑓 metric and matter thatsources the background decouple. Thus all solutions of massive gravity may be seen as 𝑀𝑓 → ∞decoupling limits of solutions of bi-gravity. This will be discussed in more depth in Section 8.4. Foran arbitrary reference metric which can be locally written as a small departures about Minkowskithe decoupling limit is derived in Eq. (8.81).

Having derived massive gravity as a consistent decoupling limit of bi-gravity, we could of coursedo the same for any multi-metric theory. For instance, out of 𝑁 -interacting fields, we could take alimit so as to decouple one of the metrics, we then obtain the theory of (𝑁 − 1)-interacting fields,all of which being massive and one decoupled massless spin-2 field.



76 Claudia de Rham

8.3 Decoupling limit of massive gravity

We now turn to a different type of decoupling limit, whose aim is to disentangle the dofs present inmassive gravity itself and analyze the ‘irrelevant interactions’ (in the usual EFT sense) that ariseat the lowest possible scale. One could naively think that such interactions arise at the scale givenby the graviton mass, but this is not so. In a generic theory of massive gravity with Fierz–Pauli atthe linear level, the first irrelevant interactions typically arise at the scale Λ5 = (𝑚4𝑀Pl)

1/5. Forthe setups we have in mind, 𝑚≪ Λ5 ≪ 𝑀Pl. But we shall see that interactions arising at such alow-energy scale are always pathological (reminiscent to the BD ghost [111, 173]), and in ghost-freemassive gravity the first (irrelevant) interactions actually arise at the scale Λ3 = (𝑚3𝑀Pl)

1/3.We start by deriving the decoupling limit in the absence of vectors (helicity-1 modes) and then

include them in the following section 8.3.4. Since we are interested in the decoupling limit aboutflat spacetime, we look at the case where Minkowski is a vacuum solution to the equations ofmotion. This is the case in the absence of a cosmological constant and a tadpole and we thus focuson the case where 𝛼0 = 𝛼1 = 0 in (6.3).

8.3.1 Interaction scales

In GR, the interactions of the helicity-2 mode arise at the very high energy scale, namely the Planckscale. In massive gravity a new scale enters and we expect some interactions to arise at a lowerenergy scale given by a geometric combination of the Planck scale and the graviton mass. Thepotential term 𝑀2

Pl𝑚2√−𝑔ℒ𝑛[𝒦[𝑔, 𝜂]] (6.3) includes generic interactions between the canonically

normalized helicity-0 (𝜋), helicity-1 (𝐴𝜇), and helicity-2 modes (ℎ𝜇𝜈) introduced in (2.48)

ℒ𝑗,𝑘,ℓ = 𝑚2𝑀2Pl

(ℎ

𝑀Pl

)𝑗 (𝜕𝐴

𝑚𝑀Pl

)2𝑘 (𝜕2𝜋

𝑚2𝑀Pl

)ℓ= Λ

−4+(𝑗+4𝑘+3ℓ)𝑗,𝑘,ℓ ℎ𝑗 (𝜕𝐴)

2𝑘 (𝜕2𝜋

)ℓ, (8.18)

at the scale

Λ𝑗,𝑘,ℓ =(𝑚2𝑘+2ℓ−2𝑀 𝑗+2𝑘+ℓ−2

Pl

)1/(𝑗+4𝑘+3ℓ−4)

, (8.19)

and with 𝑗, 𝑘, ℓ ∈ N, and 𝑗 + 2𝑘 + ℓ > 2.Clearly ,the lowest interaction scale is Λ𝑗=0,𝑘=0,ℓ=3 ≡ Λ5 = (𝑀Pl𝑚

4)1/5 which arises for anoperator of the form (𝜕2𝜋)3. If present such an interaction leads to an Ostrogradsky instabilitywhich is another manifestation of the BD ghost as identified in [173].

Even if that very interaction is absent there is actually an infinite set of dangerous interactionsof the form (𝜕2𝜋)ℓ which arise at the scale Λ𝑗=0,𝑘=0,ℓ≥3, with

Λ5 = (𝑀Pl𝑚4)1/5 ≤ (Λ𝑗=0,𝑘=0,ℓ≥3) < Λ3 = (𝑀Pl𝑚

2)1/3 . (8.20)

with Λ𝑗=0,𝑘=0,ℓ→∞ = Λ3.Any interaction with 𝑗 > 0 or 𝑘 > 0 automatically leads to a larger scale, so all the interactions

arising at a scale between Λ5 (inclusive) and Λ3 are of the form (𝜕2𝜋)ℓ and carry an Ostrogradskyinstability. For DGP we have already seen that there is no interactions at a scale below Λ3. Inwhat follows we show that same remains true for the ghost-free theory of massive gravity proposedin (6.3). To see this let us identify the interactions with 𝑗 = 𝑘 = 0 and arbitrary power ℓ for (𝜕2𝜋).

8.3.2 Operators below the scale Λ3

We now express the potential term 𝑀2Pl𝑚

2√−𝑔ℒ𝑛[𝒦] introduced in (6.3) using the metric in term

of the helicity-0 mode, where we recall that the quantity 𝒦 is defined in (6.7), as 𝒦𝜇𝜈 [𝑔, 𝑓 ] =



Massive Gravity 77

𝛿𝜇𝜈 −(√

𝑔−1𝑓)𝜇𝜈, where 𝑓 is the ‘Stuckelbergized’ reference metric given in (2.78). Since we are

interested in interactions without the helicity-2 and -1 modes (𝑗 = 𝑘 = 0), it is sufficient to followthe behaviour of the helicity-0 mode and so we have

𝑓𝜇𝜈

ℎ=𝐴=0

= 𝜂𝜇𝜈 − 2𝑀Pl𝑚2Π𝜇𝜈 +

1𝑀2

Pl𝑚4Π

2𝜇𝜈

𝑔𝜇𝜈ℎ=0

= 𝜂𝜇𝜈

⎫⎬⎭ =⇒ 𝒦𝜇𝜈 |ℎ=𝐴=0 =Π𝜇𝜈

𝑀Pl𝑚2, (8.21)

with again Π𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋 and Π2𝜇𝜈 := 𝜂𝛼𝛽Π𝜇𝛼Π𝜈𝛽 .

As a result, we infer that up to the scale Λ3 (excluded), the potential in (6.3) is

ℒmass =𝑚2𝑀2

Pl

4

√−𝑔

4∑𝑛=2

𝛼𝑛ℒ𝑛[𝒦[𝑔, 𝑓 ]]ℎ=𝐴=0

(8.22)

=𝑚2𝑀2

Pl

4

4∑𝑛=2

𝛼𝑛ℒ𝑛[

Π𝜇𝜈𝑀Pl𝑚2

](8.23)

=1

4𝜖𝜇𝜈𝛼𝛽𝜖𝜇′𝜈′𝛼′𝛽′

( 𝛼2

𝑚2𝛿𝜇

′

𝜈 𝛿𝜈′

𝜈 +𝛼3

𝑀Pl𝑚4𝛿𝜇

′

𝜈 Π𝜈′

𝜈 +𝛼4

𝑀2Pl𝑚

6Π𝜇

′

𝜈 Π𝜈′

𝜈

)Π𝛼

′

𝛼 Π𝛽′

𝛽 ,

where as mentioned earlier we focus on the case without a cosmological constant and tadpole i.e.,𝛼0 = 𝛼1 = 0. All of these interactions are total derivatives. So even though the ghost-free theoryof massive gravity does in principle involve some interactions with higher derivatives of the form(𝜕2𝜋)ℓ it does so in a very precise way so that all of these terms combine so as to give a totalderivative and being harmless.22

As a result the potential term constructed proposed in Part II (and derived from the decon-struction framework) is free of any interactions of the form (𝜕2𝜋)ℓ. This means that the BD ghostas identified in the Stuckelberg language in [173] is absent in this theory. However, at this level,the BD ghost could still reappear through different operators at the scale Λ3 or higher.

8.3.3 Λ3-decoupling limit

Since there are no operators all the way up to the scale Λ3 (excluded), we can take the decouplinglimit by sending 𝑀Pl →∞, 𝑚→ 0 and maintaining the scale Λ3 fixed.

The operators that arise at the scale Λ3 are the ones of the form (8.18) with either 𝑗 = 1, 𝑘 = 0and arbitrary ℓ ≥ 2 or with 𝑗 = 0, 𝑘 = 1 and arbitrary ℓ ≥ 1. The second case scenario leads tovector interactions of the form (𝜕𝐴)2(𝜕2𝜋)ℓ and will be studied in the next Section 8.3.4. For nowwe focus on the first kind of interactions of the form ℎ(𝜕2𝜋)ℓ,

ℒdecmass = ℎ𝜇𝜈��𝜇𝜈 , (8.24)

with [144] (see also refs. [137] and [143])

��𝜇𝜈 =𝛿

𝛿ℎ𝜇𝜈ℒmass

ℎ=𝐴=0

(8.25)

=𝑀2

Pl𝑚2

4

𝛿

𝛿ℎ𝜇𝜈

(√−𝑔

4∑𝑛=2


) ℎ=𝐴=0

.

Using the fact that

𝛿𝒦𝑛

𝛿ℎ𝜇𝜈

ℎ=𝐴=0

=𝑛

2

(Π𝑛−1𝜇𝜈 −Π𝑛𝜇𝜈

), (8.26)

22 This is actually precisely the way ghost-free massive gravity was originally constructed in [137, 144].



78 Claudia de Rham

we obtain

��𝜇𝜈 =Λ33

8

4∑𝑛=2

𝛼𝑛

(4− 𝑛Λ3𝑛3

𝑋(𝑛)𝜇𝜈 [Π] +

𝑛

Λ3(𝑛−1)3

𝑋(𝑛−1)𝜇𝜈 [Π]

), (8.27)

where the tensors 𝑋(𝑛)𝜇𝜈 are constructed out of Π𝜇𝜈 , symbolically, 𝑋(𝑛) ∼ Π(𝑛) but in such a way

that they are transverse and that their resulting equations of motion never involve more than twoderivatives on each fields,

𝑋(0)𝜇𝜇′ [𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈𝛼𝛽 (8.28)

𝑋(1)𝜇𝜇′ [𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼𝛽 𝑄

𝜈′

𝜈 (8.29)

𝑋(2)𝜇𝜇′ [𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼′𝛽 𝑄

𝜈′

𝜈 𝑄𝛼′

𝛼 (8.30)

𝑋(3)𝜇𝜇′ [𝑄] = 𝜀𝜇𝜈𝛼𝛽𝜀𝜇′𝜈′𝛼′𝛽′ 𝑄𝜈

′

𝜈 𝑄𝛼′

𝛼 𝑄𝛽′

𝛽 (8.31)

𝑋(𝑛≥4)𝜇𝜇′ [𝑄] = 0 , (8.32)

where we have included𝑋(0) and𝑋(𝑛≥4) for completeness (these become relevant for instance in thecontext of bi-gravity). The generalization of these tensors to arbitrary dimensions is straightforwardand in 𝑑-spacetime dimensions there are 𝑑 such tensors, symbolically 𝑋(𝑛) = 𝜀𝜀Π𝑛𝛿𝑑−𝑛−1 for𝑛 = 0, · · · , 𝑑− 1.

Since we are dealing with the decoupling limit with 𝑀Pl → ∞ the metric is flat 𝑔𝜇𝜈 = 𝜂𝜇𝜈 +𝑀−1

Pl ℎ𝜇𝜈 → 𝜂𝜇𝜈 and all indices are raised and lowered with respect to the Minkowski metric. These

tensors 𝑋(𝑛)𝜇𝜈 can be written more explicitly as follows

𝑋(0)𝜇𝜈 [𝑄] = 3!𝜂𝜇𝜈 (8.33)

𝑋(1)𝜇𝜈 [𝑄] = 2! ([𝑄]𝜂𝜇𝜈 −𝑄𝜇𝜈) (8.34)

𝑋(2)𝜇𝜈 [𝑄] = ([𝑄]2 − [𝑄2])𝜂𝜇𝜈 − 2([𝑄]𝑄𝜇𝜈 −𝑄2

𝜇𝜈) (8.35)

𝑋(3)𝜇𝜈 [𝑄] = ([𝑄]3 − 3[𝑄][𝑄2] + 2[𝑄3])𝜂𝜇𝜈 (8.36)

−3([𝑄]2𝑄𝜇𝜈 − 2[𝑄]𝑄2

𝜇𝜈 − [𝑄2]𝑄𝜇𝜈 + 2𝑄3𝜇𝜈

).

Note that they also satisfy the recursive relation

𝑋(𝑛)𝜇𝜈 =

1

4− 𝑛(−𝑛Π𝛼𝜇𝛿𝛽𝜈 +Π𝛼𝛽𝜂𝜇𝜈

)𝑋

(𝑛−1)𝛼𝛽 , (8.37)

with 𝑋(0)𝜇𝜈 = 3!𝜂𝜇𝜈 .

Decoupling limit

From the expression of these tensors 𝑋𝜇𝜈 in terms of the fully antisymmetric Levi-Cevita tensors,

it is clear that the tensors 𝑋(𝑛)𝜇𝜈 are transverse and that the equations of motion of ℎ𝜇𝜈��𝜇𝜈 with

respect to both ℎ and 𝜋 never involve more than two derivatives. This decoupling limit is thusfree of the Ostrogradsky instability which is the way the BD ghost would manifest itself in thislanguage. This decoupling limit is actually free of any ghost-lie instability and the whole theory isfree of the BD even beyond the decoupling limit as we shall see in depth in Section 7.

Not only does the potential term proposed in (6.3) remove any potential interactions of theform (𝜕2𝜋)ℓ which could have arisen at an energy between Λ5 = (𝑀Pl𝑚

4)1/5 and Λ3 , but it alsoensures that the interactions that arise at the scale Λ3 are healthy.



Massive Gravity 79

As already mentioned, in the decoupling limit 𝑀Pl →∞ the metric reduces to Minkowski andthe standard Einstein–Hilbert term simply reduces to its linearized version. As a result, neglectingthe vectors for now the full Λ3-decoupling limit of ghost-free massive gravity is given by

ℒΛ3= −1

4ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 +

1

8ℎ𝜇𝜈

(2𝛼2𝑋

(1)𝜇𝜈 +

2𝛼2 + 3𝛼3

Λ33

𝑋(2)𝜇𝜈 +

𝛼3 + 4𝛼4

Λ63

𝑋(3)𝜇𝜈

)(8.38)

= −1

4ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 + ℎ𝜇𝜈

3∑𝑛=1

𝑎𝑛

Λ3(𝑛−1)3

𝑋(𝑛)𝜇𝜈 ,

with 𝑎1 = 𝛼2/4, 𝑎2 = (2𝛼2 + 3𝛼3)/8 and 𝑎3 = (𝛼3 + 4𝛼4)/8 and the correct normalization shouldbe 𝛼2 = 1.

Unmixing and Galileons

As was already the case at the linearized level for the Fierz–Pauli theory (see Eqs. (2.47) and(2.48)) the kinetic term for the helicity-0 mode appears mixed with the helicity-2 mode. It is thusconvenient to diagonalize these two modes by performing the following shift,

ℎ𝜇𝜈 = ℎ𝜇𝜈 + 𝛼2𝜋𝜂𝜇𝜈 −2𝛼2 + 3𝛼3

2Λ33

𝜕𝜇𝜋𝜕𝜈𝜋 , (8.39)

where the non-linear term has been included to unmix the coupling ℎ𝜇𝜈𝑋(2)𝜇𝜈 , leading to the fol-

lowing decoupling limit [137]

ℒΛ3= −1

4

[ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 +

5∑𝑛=2

𝑐𝑛

Λ3(𝑛−2)3

ℒ(𝑛)(Gal)[𝜋]−

2(𝛼3 + 4𝛼4)

Λ63

ℎ𝜇𝜈𝑋(3)𝜇𝜈

], (8.40)

where we introduced the Galileon Lagrangians ℒ(𝑛)(Gal)[𝜋] as defined in Ref. [412]

ℒ(𝑛)(Gal)[𝜋] =

1

(6− 𝑛)!(𝜕𝜋)2ℒ𝑛−2[Π] (8.41)

= − 2

𝑛(5− 𝑛)!𝜋ℒ𝑛−1[Π] , (8.42)

where the Lagrangians ℒ𝑛[𝑄] = 𝜀𝜀𝑄𝑛𝛿4−𝑛 for a tensor 𝑄𝜇𝜈 are defined in (6.9) – (6.13), or moreexplicitly in (6.14) – (6.18), leading to the explicit form for the Galileon Lagrangians

ℒ(2)(Gal)[𝜋] = (𝜕𝜋)2 (8.43)

ℒ(3)(Gal)[𝜋] = (𝜕𝜋)2[Π] (8.44)

ℒ(4)(Gal)[𝜋] = (𝜕𝜋)2

([Π]2 − [Π2]

)(8.45)

ℒ(5)(Gal)[𝜋] = (𝜕𝜋)2

([Π]3 − 3[Π][Π2] + 2[Π3]

), (8.46)

and the coefficients 𝑐𝑛 are given in terms of the 𝛼𝑛 as follows,

𝑐2 = 3𝛼22 , 𝑐3 = 3

2𝛼2(2𝛼2 + 3𝛼3) ,

𝑐4 = 14 (4𝛼

22 + 9𝛼2

3 + 16𝛼2(𝛼3 + 𝛼4)) , 𝑐5 = 58 (2𝛼2 + 3𝛼3)(𝛼3 + 4𝛼4) .

(8.47)

Setting 𝛼2 = 1, we indeed recover the same normalization of −3/4(𝜕𝜋)2 for the helicity-0 modefound in (2.48).



80 Claudia de Rham

𝑋(3)-coupling

In general, the last coupling ℎ𝜇𝜈𝑋(3)𝜇𝜈 between the helicity-2 and helicity-0 mode cannot be removed

by a local field redefinition. The non-local field redefinition

ℎ𝜇𝜈 → ℎ𝜇𝜈 +𝐺massless𝜇𝜈𝛼𝛽 𝑋(3)𝛼𝛽 , (8.48)

where 𝐺massless𝜇𝜈𝛼𝛽 is the propagator for a massless spin-2 field as defined in (2.64), fully diagonalizes

the helicity-0 and -2 mode at the price of introducing non-local interactions for 𝜋.Note however that these non-local interactions do not hide any new degrees of freedom. Fur-

thermore, about some specific backgrounds, the field redefinition is local. Indeed focusing on staticand spherically symmetric configurations if we consider 𝜋 = 𝜋0(𝑟) and ℎ𝜇𝜈 given by

ℎ𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −𝜓(𝑟) d𝑡2 + 𝜑(𝑟) d𝑟2 , (8.49)

so thatℎ𝜇𝜈𝑋(3)

𝜇𝜈 = −𝜓′(𝑟)𝜋′0(𝑟)

3 . (8.50)

The standard kinetic term for 𝜓 sets 𝜓′(𝑟) = 𝜑(𝑟)/𝑟 as in GR and the 𝑋(3) coupling can beabsorbed via the field redefinition, 𝜑→ 𝜑− 2(𝛼3 + 4𝛼4)𝜋

′0(𝑟)

3/𝑟Λ−63 , leading to the following new

sextic interactions for 𝜋,

ℎ𝜇𝜈𝑋(3)𝜇𝜈 → −

1

𝑟2𝜋′0(𝑟)

6 , (8.51)

interestingly this new order-6 term satisfy all the relations of a Galileon interaction but cannot beexpressed covariantly in a local way. See [61] for more details on spherically symmetric configura-tions with the 𝑋(3)-coupling.

8.3.4 Vector interactions in the Λ3-decoupling limit

As can be seen from the relation (8.19), the scale associated with interactions mixing two helicity-1fields with an arbitrary number of fields 𝜋, (𝑗 = 0, 𝑘 = 1 and arbitrary ℓ) is also Λ3. So at thatscale, there are actually an infinite number of interactions when including the mixing with betweenthe helicity-1 and -0 modes (however as mentioned previously, since the vector field always appearsquadratically it is always consistent to set them to zero as was performed previously).

The full decoupling limit including these interactions has been derived in Ref. [419], (see alsoRef. [238]) using the vielbein formulation of massive gravity as in (6.1) and we review the formalismand the results in what follows.

In addition to the Stuckelberg fields associated with local covariance, in the vielbein formulationone also needs to introduce 6 additional Stuckelberg fields 𝜔𝑎𝑏 associated to local Lorentz invariance,𝜔𝑎𝑏 = −𝜔𝑏𝑎. These are non-dynamical since they never appear with derivatives, and can thusbe treated as auxiliary fields which can be integrated. It is however useful to keep them in thedecoupling limit action, so as to retain a closes-form expression. In terms of the Lorentz Stuckelbergfields, the full decoupling limit of massive gravity in four dimensions at the scale Λ3 is then (beforediagonalization) [419]

ℒ(0)Λ3

= −1


1

2ℎ𝜇𝜈

3∑𝑛=1

𝑎𝑛

Λ3(𝑛−1)3

𝑋(𝑛)𝜇𝜈 (8.52)

+3𝛽18𝛿𝛼𝛽𝛾𝛿𝑎𝑏𝑐𝑑 𝛿

𝑎𝛼

(𝛿𝑏𝛽𝐹

𝑐𝛾𝜔

𝑑𝛿 + 2[𝜔𝑏𝛽𝜔

𝑐𝛾 +

1

2𝛿𝑏𝛽𝜔

𝑐𝜇𝜔

𝜇𝛾 ](𝛿 +Π)𝑑𝛿

)+𝛽28𝛿𝛼𝛽𝛾𝛿𝑎𝑏𝑐𝑑 (𝛿 +Π)

𝑎𝛼

(2𝛿𝑏𝛽𝐹

𝑐𝛾𝜔

𝑑𝛿 + [𝜔𝑏𝛽𝜔

𝑐𝛾 + 𝛿𝑏𝛽𝜔

𝑐𝜇𝜔

𝜇𝛾 ](𝛿 +Π)𝑑𝛿

)+𝛽348𝛿𝛼𝛽𝛾𝛿𝑎𝑏𝑐𝑑 (𝛿 +Π)

𝑎𝛼 (𝛿 +Π)

𝑏𝛽

(3𝐹 𝑐𝛾𝜔

𝑑𝛿 + 𝜔𝑐𝜇𝜔

𝜇𝛾(𝛿 +Π)𝑑𝛿

),



Massive Gravity 81

(the superscript (0) indicates that this decoupling limit is taken with Minkowski as a referencemetric), with 𝐹𝑎𝑏 = 𝜕𝑎𝐴𝑏 − 𝜕𝑏𝐴𝑎 and the coefficients 𝛽𝑛 are related to the 𝛼𝑛 as in (6.28).

The auxiliary Lorentz Stuckelberg fields carries all the non-linear mixing between the helicity-0and -1 modes,

𝜔𝑎𝑏 =

∫ ∞

0

d𝑢 𝑒−2𝑢𝑒−𝑢Π𝑎′𝑎 𝐹𝑎′𝑏′𝑒

−𝑢Π𝑏′𝑏 (8.53)

=∑𝑛,𝑚

(𝑛+𝑚)!

21+𝑛+𝑚𝑛!𝑚!(−1)𝑛+𝑚 (Π𝑛 𝐹 Π𝑚)𝑎𝑏 . (8.54)

In some special cases these sets of interactions can be resummed exactly, as was first performedin [139], (see also Refs. [364, 456]).

This decoupling limit includes non-linear combinations of the second-derivative tensor Π𝜇𝜈 andthe first derivative Maxwell tensor 𝐹𝜇𝜈 . Nevertheless, the structure of the interactions is gaugeinvariant for 𝐴𝜇, and there are no higher derivatives on 𝐴 in the equation of motion for 𝐴, sothe equations of motions for both the helicity-1 and -2 modes are manifestly second order andpropagating the correct degrees of freedom. The situation is more subtle for the helicity-0 mode.Taking the equation of motion for that field would lead to higher derivatives on 𝜋 itself as well ason the helicity-1 field. Since this theory has been proven to be ghost-free by different means (seeSection 7), it must be that the higher derivatives in that equation are nothing else but the derivativeof the equation of motion for the helicity-1 mode similarly as what happens in Section 7.2.

When working beyond the decoupling limit, the even the equation of motion with respect tothe helicity-1 mode is no longer manifestly well-behaved, but as we shall see below, the Stuckelbergfields are no longer the correct representation of the physical degrees of freedom. As we shall seebelow, the proper number of degrees of freedom is nonetheless maintained when working beyondthe decoupling limit.

8.3.5 Beyond the decoupling limit

Physical degrees of freedom

In Section 8.3, we have introduced four Stuckelberg fields 𝜑𝑎 which transform as scalar fields undercoordinate transformation, so that the action of massive gravity is invariant under coordinatetransformations. Furthermore, the action is also invariant under global Lorentz transformations inthe field space,

𝑥𝜇 → 𝑥𝜇 , 𝑔𝜇𝜈 → 𝑔𝜇𝜈 , and 𝜑𝑎 → Λ𝑎𝑏𝜑𝑏 . (8.55)

In the DL, taking 𝑀Pl → ∞, all fields are living on flat space-time, so in that limit, there is anadditional global Lorentz symmetry acting this time on the space-time,

𝑥𝜇 → Λ𝜇𝜈 𝑥𝜈 , ℎ𝜇𝜈 → Λ𝛼𝜇Λ

𝛽𝜈ℎ𝛼𝛽 , and 𝜑𝑎 → 𝜑𝑎 . (8.56)

The internal and space-time Lorentz symmetries are independent, (the internal one is alwayspresent while the space-time one is only there in the DL). In the DL we can identify both groupsand work in the representation of the single group, so that the action is invariant under,

𝑥𝜇 → Λ𝜇𝜈 𝑥𝜈 , ℎ𝜇𝜈 → Λ𝛼𝜇Λ

𝛽𝜈ℎ𝛼𝛽 , and 𝜑𝑎 → Λ𝑎𝑏𝜑

𝑏 . (8.57)

The Stuckelberg fields 𝜑𝑎 then behave as Lorentz vectors under this identified group, and 𝜋 definedpreviously behaves as a Lorentz scalar. The helicity-0 mode of the graviton also behaves as a scalarin this limit, and 𝜋 captures the behavior of the graviton helicity-0 mode. So in the DL limit, theright requirement for the absence of BD ghost is indeed the requirement that the equations of



82 Claudia de Rham

motion for 𝜋 remain at most second order (time) in derivative as was pointed out in [173], (seealso [111]). However, beyond the DL, the helicity-0 mode of the graviton does not behave as ascalar field and neither does the 𝜋 in the split of the Stuckelberg fields. So beyond the DL thereis no reason to anticipate that 𝜋 captures a whole degree of freedom, and it indeed, it does not.Beyond the DL, the equation of motion for 𝜋 will typically involve higher derivatives, but thecorrect requirement for the absence of ghost is different, as explained in Section 7.2. One shouldinstead go back to the original four scalar Stuckelberg fields 𝜑𝑎 and check that out of these fourfields only three of them be dynamical. This has been shown to be the case in Section 7.2. Thesethree degrees of freedom, together with the two standard graviton polarizations then gives thecorrect five degrees of freedom and circumvent the BD ghost.

Recently, much progress has been made in deriving the decoupling limit about arbitrary back-grounds, see Ref. [369].

8.3.6 Decoupling limit on (Anti) de Sitter

Linearized theory and Higuchi bound

Before deriving the decoupling limit of massive gravity on (Anti) de Sitter, we first need to analyzethe linearized theory so as to infer the proper canonical normalization of the propagating dofs andthe proper scaling in the decoupling limit, similarly as what was performed for massive gravitywith flat reference metric. For simplicity we focus on (3 + 1) dimensions here, and when relevantgive the result in arbitrary dimensions. Linearized massive gravity on (A)dS was first derivedin [307, 308]. Since we are concerned with the decoupling limit of ghost-free massive gravity, wefollow in this section the procedure presented in [154]. We also focus on the dS case first beforecommenting on the extension to AdS.

At the linearized level about dS, ghost-free massive gravity reduces to the Fierz–Pauli actionwith 𝑔𝜇𝜈 = 𝛾𝜇𝜈+ℎ𝜇𝜈 = 𝛾𝜇𝜈+ℎ𝜇𝜈/𝑀Pl, where 𝛾𝜇𝜈 is the dS metric with constant Hubble parameter𝐻0,

ℒ(2)MG, dS = −1

4ℎ𝜇𝜈(ℰdS)𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 −

𝑚2

8𝛾𝜇𝜈𝛾𝛼𝛽 (𝐻𝜇𝛼𝐻𝜈𝛽 −𝐻𝜇𝜈𝐻𝛼𝛽) , (8.58)

where 𝐻𝜇𝜈 is the tensor fluctuation as introduced in (2.80), although now considered about the dSmetric,

𝐻𝜇𝜈 = ℎ𝜇𝜈 + 2∇(𝜇𝐴𝜈)

𝑚+ 2

Π𝜇𝜈𝑚2

(8.59)

− 1

𝑀Pl

[∇𝜇𝐴𝛼𝑚

+Π𝜇𝛼𝑚2

] [∇𝜈𝐴𝛽𝑚

+Π𝜈𝛽𝑚2

]𝛾𝛼𝛽 ,

with Π𝜇𝜈 = ∇𝜇∇𝜈𝜋, ∇ being the covariant derivative with respect to the dS metric 𝛾𝜇𝜈 and indices

are raised and lowered with respect to this same metric. Similarly, ℰdS is now the Lichnerowiczoperator on de Sitter,

(ℰdS)𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 = −1

2

[2ℎ𝜇𝜈 − 2∇(𝜇∇𝛼ℎ𝛼𝜈) +∇𝜇∇𝜈ℎ (8.60)

−𝛾𝜇𝜈(2ℎ−∇𝛼∇𝛽ℎ𝛼𝛽) + 6𝐻20

(ℎ𝜇𝜈 −

1

2ℎ𝛾𝜇𝜈

)].

So at the linearized level and neglecting the vector fields, the helicity-0 and -2 mode of massivegravity on dS behave as

ℒ(2)MG, dS = −1


𝑚2

8


)− 1

8𝐹 2𝜇𝜈 (8.61)

−1

2ℎ𝜇𝜈 (Π𝜇𝜈 − [Π]𝛾𝜇𝜈)−

1

2𝑚2

([Π2]− [Π]2

).



Massive Gravity 83

After integration by parts, [Π2] = [Π]2 − 3𝐻2(𝜕𝜋)2. The helicity-2 and -0 modes are thus diago-nalized as in flat space-time by setting ℎ𝜇𝜈 = ℎ𝜇𝜈 + 𝜋𝛾𝜇𝜈 ,

ℒ(2)MG, dS = −1


𝑚2

8


)− 1

8𝐹 2𝜇𝜈 (8.62)

−3

4

(1− 2

(𝐻

𝑚

)2)(

(𝜕𝜋)2 −𝑚2ℎ𝜋 − 2𝑚2𝜋2

).

The most important difference from linearized massive gravity on Minkowski is that the properlycanonically normalized helicity-0 mode is now instead

𝜑 =

√1− 2

𝐻2

𝑚2𝜋 . (8.63)

For a standard coupling of the form 1𝑀Pl

𝜋𝑇 , where 𝑇 is the trace of the stress-energy tensor, as we

would infer from the coupling 1𝑀Pl

ℎ𝜇𝜈𝑇𝜇𝜈 after the shift ℎ𝜇𝜈 = ℎ𝜇𝜈 + 𝜋𝛾𝜇𝜈 , this means that the

properly normalized helicity-0 mode couples as

ℒmatterhelicity−0 =

𝑚2

𝑀Pl

√𝑚2 − 2𝐻2

𝜑𝑇 , (8.64)

and that coupling vanishes in the massless limit. This might suggest that in the massless limit𝑚 → 0, the helicity-0 mode decouples, which would imply the absence of the standard vDVZdiscontinuity on (Anti) de Sitter [358, 430], unlike what was found on Minkowski, see Section 2.2.3,which confirms the Newtonian approximation presented in [186].

While this observation is correct on AdS, in the dS one cannot take the massless limit withoutsimultaneously sending 𝐻 → 0 at least the same rate. As a result, it would be incorrect to deducethat the helicity-0 mode decouples in the massless limit of massive gravity on dS.

To be more precise, the linearized action (8.62) is free from ghost and tachyons only if 𝑚 ≡ 0which corresponds to GR, or if𝑚2 > 2𝐻2, which corresponds to the well-know Higuchi bound [307,190]. In 𝑑 spacetime dimensions, the Higuchi bound is 𝑚2 > (𝑑 − 2)𝐻2. In other words, on dSthere is a forbidden range for the graviton mass, a theory with 0 < 𝑚2 < 2𝐻2 or with 𝑚2 < 0always excites at least one ghost degree of freedom. Notice that this ghost, (which we shall referto as the Higuchi ghost from now on) is distinct from the BD ghost which corresponded to anadditional sixth degree of freedom. Here the theory propagates five dof (in four dimensions) andis thus free from the BD ghost (at least at this level), but at least one of the five dofs is a ghost.When 0 < 𝑚2 < 2𝐻2, the ghost is the helicity-0 mode, while for 𝑚2 < 0, the ghost is he helicity-1

mode (at quadratic order the helicity-1 mode comes in as −𝑚2

4 𝐹2𝜇𝜈). Furthermore, when 𝑚2 < 0,

both the helicity-2 and -0 are also tachyonic, although this is arguably not necessarily a severeproblem, especially not if the graviton mass is of the order of the Hubble parameter today, asit would take an amount of time comparable to the age of the Universe to see the effect of thistachyonic behavior. Finally, the case 𝑚2 = 2𝐻2 (or 𝑚2 = (𝑑 − 2)𝐻2 in 𝑑 spacetime dimensions),represents the partially massless case where the helicity-0 mode disappears. As we shall see inSection 9.3, this is nothing other than a linear artefact and non-linearly the helicity-0 mode alwaysreappears, so the PM case is infinitely strongly coupled and always pathological.

A summary of the different bounds is provided below as well as in Figure 4:

∙ 𝑚2 < 0: Helicity-1 modes are ghost, helicity-2 and -0 are tachyonic, sick theory

∙ 𝑚2 = 0: General Relativity: two healthy (helicity-2) degrees of freedom, healthy theory,

∙ 0 < 𝑚2 < 2𝐻2: One “Higuchi ghost” (helicity-0 mode) and four healthy degrees of freedom(helicity-2 and -1 modes), sick theory,



84 Claudia de Rham

∙ 𝑚2 = 2𝐻2: Partially Massless Gravity: Four healthy degrees (helicity-2 and -1 modes),and one infinitely strongly coupled dof (helicity-0 mode), sick theory,

∙ 𝑚2 > 2𝐻2: Massive Gravity on dS: Five healthy degrees of freedom, healthy theory.

Figure 4: Degrees of freedom for massive gravity on a maximally symmetric reference metric. The onlytheoretically allowed regions are the upper left green region and the line 𝑚 = 0 corresponding to GR.

Massless and decoupling limit

∙ As one can see from Figure 4, in the case where 𝐻2 < 0 (corresponding to massive gravityon AdS), one can take the massless limit 𝑚 → 0 while keeping the AdS length scale fixedin that limit. In that limit, the helicity-0 mode decouples from external matter sources andthere is no vDVZ discontinuity. Notice however that the helicity-0 mode is nevertheless stillstrongly coupled at a low energy scale.

When considering the decoupling limit 𝑚 → 0, 𝑀Pl → ∞ of massive gravity on AdS, wehave the choice on how we treat the scale 𝐻 in that limit. Keeping the AdS length scalefixed in that limit could lead to an interesting phenomenology in its own right, but is yet tobe explored in depth.

∙ In the dS case, the Higuchi forbidden region prevents us from taking the massless limitwhile keeping the scale 𝐻 fixed. As a result, the massless limit is only consistent if 𝐻 → 0



Massive Gravity 85

simultaneously as 𝑚 → 0 and we thus recover the vDVZ discontinuity at the linear level inthat limit.

When considering the decoupling limit 𝑚→ 0, 𝑀Pl →∞ of massive gravity on dS, we alsohave to send 𝐻 → 0. If 𝐻/𝑚 → 0 in that limit, we then recover the same decoupling limitas for massive gravity on Minkowski, and all the results of Section 8.3 apply. The case ofinterest is thus when the ratio 𝐻/𝑚 remains fixed in the decoupling limit.

Decoupling limit

When taking the decoupling limit of massive gravity on dS, there are two additional contributionsto take into account:

∙ First, as mentioned in Section 8.3.5, care needs to be applied to properly identify the helicity-0mode on a curved background. In the case of (A)dS, the formalism was provided in Ref. [154]by embedding a 𝑑-dimensional de Sitter spacetime into a flat (𝑑+ 1)-dimensional spacetimewhere the standard Stuckelberg trick could be applied. As a result the ‘covariant’ fluctuationdefined in (2.80) and used in (8.59) needs to be generalized to (see Ref. [154] for details)

1

𝑀Pl𝐻𝜇𝜈 =

1

𝑀Plℎ𝜇𝜈 +

2

Λ33

Π𝜇𝜈 −1

Λ63

Π2𝜇𝜈 (8.65)

+1

Λ33

𝐻2

𝑚2

((𝜕𝜋)2(𝛾𝜇𝜈 −

2

Λ33

Π𝜇𝜈)−1

Λ63

Π𝜇𝛼Π𝜈𝛽𝜕𝛼𝜋𝜕𝛽𝜋

)+𝐻2𝐻

2

𝑚2

(𝜕𝜋)4

Λ93

+ · · · .

Any corrections in the third line vanish in the decoupling limit and can thus be ignored, butthe corrections of order 𝐻2 in the second line lead to new non-trivial contributions.

∙ Second, as already encountered at the linearized level, what were total derivatives in Minkowski(for instance the combination [Π2]− [Π]2), now lead to new contributions on de Sitter. Afterintegration by parts, 𝑚−2([Π2] − [Π]2) = 𝑚−2𝑅𝜇𝜈𝜕

𝜇𝜋𝜕𝜈𝜋 = 12𝐻2/𝑚2(𝜕𝜋)2. This was theorigin of the new kinetic structure for massive gravity on de Sitter and will have furthereffects in the decoupling limit when considering similar contributions from ℒ3,4(Π), whereℒ3,4 are defined in (6.12, 6.13) or more explicitly in (6.17, 6.18).

Taking these two effects into account, we obtain the full decoupling limit for massive gravity on deSitter,

ℒ(dS)Λ3

= ℒ(0)Λ3

+𝐻2

𝑚2

5∑𝑛=2

𝜆𝑛

Λ3(𝑛−1)3

ℒ(𝑛)(Gal)[𝜋] , (8.66)

where ℒ(0)Λ3

is the full Lagrangian obtained in the decoupling limit in Minkowski and given in (8.52),

and ℒ(𝑛)(Gal) are the Galileon Lagrangians as encountered previously. Notice that while the ratio

𝐻/𝑚 remains fixed,this decoupling limit is taken with 𝐻,𝑚→ 0, so all the fields in (8.66) live ona Minkowski metric. The constant coefficients 𝜆𝑛 depend on the free parameters of the ghost-freetheory of massive gravity, for the theory (6.3) with 𝛼1 = 0 and 𝛼2 = 1, we have

𝜆2 =3

2, 𝜆3 =

3

4(1 + 2𝛼3) , 𝜆4 =

1

4(−1 + 6𝛼4) , 𝜆5 = − 3

16(𝛼3 + 4𝛼4) . (8.67)



86 Claudia de Rham

At this point we may perform the same field redefinition (8.39) as in flat space and obtain thefollowing semi-diagonalized decoupling limit,

ℒdS)Λ3

= −1


𝛼3 + 4𝛼4

8Λ93

ℎ𝜇𝜈𝑋(3)𝜇𝜈 +

5∑𝑛=2

𝑐𝑛

Λ3(𝑛−2)3

ℒ(𝑛)(Gal)[𝜋] (8.68)

+ Contributions from the helicity-1 modes ,

where the contributions from the helicity-1 modes are the same as the ones provided in (8.52),and the new coefficients 𝑐𝑛 = −𝑐𝑛/4 +𝐻2/𝑚2𝜆𝑛 cancel identically for 𝑚2 = 2𝐻2, 𝛼3 = −1 and𝛼4 = −𝛼3/4 = 1/4, as pointed out in [154], and the same result holds for bi-gravity as pointed outin [301]. Interestingly, for these specific parameters, the helicity-0 loses its kinetic term, and anyself-mixing as well as any mixing with the helicity-2 mode. Nevertheless, the mixing between thehelicity-1 and -0 mode as presented in (8.52) are still alive. There are no choices of parameterswhich would allow to remove the mixing with the helicity-1 mode and as a result, the helicity-0mode generically reappears through that mixing. The loss of its kinetic term implies that the fieldis infinitely strongly coupled on a configuration with zero vev for the helicity-1 mode and is thusan ill-defined theory. This was confirmed in various independent studies, see Refs. [185, 147].

8.4 Λ3-decoupling limit of bi-gravity

We now proceed to derive the Λ3-decoupling limit of bi-gravity, and we will see how to recoverthe decoupling limit about any reference metric (including Minkowski and de Sitter) as specialcases. As already seen in Section 8.3.4, the full DL is better formulated in the vielbein language,even though in that case Stuckelberg fields ought to be introduced for the broken diff and thebroken Lorentz. Yet, this is a small price to pay, to keep the action in a much simpler form. Wethus proceed in the rest of this section by deriving the Λ3-decoupling of bi-gravity and start in itsvielbein formulation. We follow the derivation and formulation presented in [224]. As previously,we focus on (3 + 1)-spacetime dimensions, although the whole formalism is trivially generalizableto arbitrary dimensions.

We start with the action (5.43) for bi-gravity, with the interaction

ℒ𝑔,𝑓 =𝑀2

Pl𝑚2

4

∫d4𝑥√−𝑔

4∑𝑛=0

𝛼𝑛ℒ𝑛[𝒦[𝑔, 𝑓 ]] (8.69)

= −𝑀2Pl𝑚

2

2𝜀𝑎𝑏𝑐𝑑

∫ [𝛽04!𝑒𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 + 𝛽1

3!𝑓𝑎 ∧ 𝑒𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 (8.70)

+𝛽22!2!

𝑓𝑎 ∧ 𝑓 𝑏 ∧ 𝑒𝑐 ∧ 𝑒𝑑 + 𝛽33!𝑓𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑒𝑑 + 𝛽4

4!𝑓𝑎 ∧ 𝑓 𝑏 ∧ 𝑓 𝑐 ∧ 𝑓𝑑

],

where the relation between the 𝛼’s and the 𝛽’s is given in (6.28).We now introduce Stuckelberg fields 𝜑𝑎 = 𝑥𝑎 − 𝜒𝑎 for diffs and Λ𝑎𝑏 for the local Lorentz. In

the case of massive gravity, there was no ambiguity in how to perform this ‘Stuckelbergization’ butin the case of bi-gravity, one can either ‘Stuckelbergize the metric 𝑓𝜇𝜈 or the metric 𝑔𝜇𝜈 . In otherwords the broken diffs and local Lorentz symmetries can be restored by performing either one ofthe two replacements in (8.69),

𝑓𝑎𝜇 → 𝑓𝑎𝜇 = Λ𝑎𝑏𝑓𝑏𝑐 (𝜑(𝑥)) 𝜕𝜇𝜑

𝑐 . (8.71)

or alternatively

𝑒𝑎𝜇 → 𝑒𝑎𝜇 = Λ𝑎𝑏𝑒𝑏𝑐(𝜑(𝑥)) 𝜕𝜇𝜑

𝑐 . (8.72)



Massive Gravity 87

For now we stick to the first choice (8.71) but keep in mind that this freedom has deep consequencesfor the theory, and is at the origin of the duality presented in Section 10.7.

Since we are interested in the decoupling limit, we now perform the following splits, (seeRef. [419] for more details),

𝑒𝑎𝜇 = 𝑒𝑎𝜇 +1

2𝑀Plℎ𝑎𝜇 , 𝑓𝑎𝜇 = 𝑒𝑎𝜇 +

1

2𝑀𝑓𝑣𝑎𝜇

Λ𝑎𝑏 = 𝑒��𝑎𝑏 = 𝛿𝑎𝑏 + ��𝑎𝑏 +

1

2��𝑎𝑐��

𝑐𝑏 + · · ·

��𝑎𝑏 =𝜔𝑎𝑏𝑚𝑀Pl

𝜕𝜇𝜑𝑎 = 𝜕𝜇

(𝑥𝑎 +

𝐴𝑎

𝑚𝑀Pl+𝜕𝑎𝜋

Λ33

)(8.73)

and perform the scaling or decoupling limit,

𝑀Pl →∞ , 𝑀𝑓 →∞ , 𝑚→ 0 (8.74)

while keeping

Λ3 = (𝑚2𝑀Pl)13 → constant , 𝑀Pl/𝑀𝑓 → constant , (8.75)

and 𝛽𝑛 → constant .

Before performing any change of variables (any diagonalization), in addition to the kinetic termfor quadratic ℎ, 𝑣 and 𝐴, there are three contributions to the decoupling limit of bi-gravity:

¶ Mixing of the helicity-0 mode with the helicity-1 mode 𝐴𝜇, as derived in (8.52),

· Mixing of the helicity-0 mode with the helicity-2 mode ℎ𝑎𝜇, as derived in (8.40),

¸ Mixing of the helicity-0 mode with the new helicity-2 mode 𝑣𝑎𝜇,

noticing that before field redefinitions, the helicity-0 mode do not self-interact (their self-interactions are constructed so as to be total derivatives).

As already explained in Section 8.3.6, the first contribution ¶ arising from the mixing betweenthe helicity-0 and -1 modes is the same (in the decoupling limit) as what was obtained in Minkowski(and is independent of the coefficients 𝛽𝑛 or 𝛼𝑛). This implies that the can be directly read of fromthe three last lines of (8.52). These contributions are the most complicated parts of the decouplinglimit but remained unaffected by the dynamics of 𝑣, i.e., unaffected by the bi-gravity nature of thetheory. This statement simply follows from scaling considerations. In the decoupling limit therecannot be any mixing between the helicity-1 and neither of the two helicity-2 modes. As a result,the helicity-1 modes only mix with themselves and the helicity-0 mode. Hence, in the scaling limit(8.74, 8.75) the helicity-1 decouples from the massless spin-2 field.

Furthermore, the first line of (8.52) which corresponds to the dynamics of ℎ𝑎𝜇 and the helicity-0mode is also unaffected by the bi-gravity nature of the theory. Hence, the second contribution ·is the also the same as previously derived. As a result, the only new ingredient in bi-gravity is themixing ¸ between the helicity-0 mode and the second helicity-2 mode 𝑣𝑎𝜇, given by a fixing of theform ℎ𝜇𝜈𝑋𝜇𝜈 .

Unsurprisingly, these new contributions have the same form as ·, with three distinctions: Firstthe way the coefficients enter in the expressions get modified ever so slightly (𝛽1 → 𝛽1/3 and𝛽3 → 3𝛽3). Second, in the mass term the space-time index for 𝑣𝑎𝜇 ought to dressed with theStuckelberg field,

𝑣𝑎𝜇 → 𝑣𝑎𝑏 𝜕𝜇𝜑𝑏 = 𝑣𝑎𝑏 (𝛿

𝑏𝜇 +Π𝑏𝜇/Λ

33) . (8.76)



88 Claudia de Rham

Finally, and most importantly, the helicity-2 field 𝑣𝜇𝑎 (which enters in the mass term) is now afunction of the ‘Stuckelbergized’ coordinates 𝜑𝑎, which in the decoupling limit means that for themass term

𝑣𝑎𝑏 = 𝑣𝑎𝑏 [𝑥𝜇 + 𝜕𝜇𝜋/Λ3

3] ≡ 𝑣𝑎𝑏 [��] . (8.77)

These two effects do not need to be taken into account for the 𝑣 that enters in its standard curvatureterm as it is Lorentz and diff invariant.

Taking these three considerations into account, one obtains the decoupling limit for bi-gravity,

ℒ(bi−gravity)Λ3

= ℒ(0)Λ3− 1

4𝑣𝜇𝜈 [𝑥]ℰ𝛼𝛽𝜇𝜈 𝑣𝛼𝛽 [𝑥] (8.78)

− 1

2

𝑀Pl

𝑀𝑓𝑣𝜇𝛽 [��]

(𝛿𝜈𝛽 +

Π𝜈𝛽Λ33

) 3∑𝑛=0

𝛽𝑛+1

Λ3(𝑛−1)3

𝑋(𝑛)𝜇𝜈 [Π] ,

with 𝛽𝑛 = 𝛽𝑛/(4−𝑛)!(𝑛−1)!. Modulo the non-trivial dependence on the coordinate �� = 𝑥+𝜕𝜋/Λ33,

this is a remarkable simple decoupling limit for bi-gravity. Out of this decoupling limit we canre-derive all the DL found previously very elegantly.

Notice as well the presence of a tadpole for 𝑣 if 𝛽1 = 0. When this tadpole vanishes (as well asthe one for ℎ), one can further take the limit𝑀𝑓 →∞ keeping all the other 𝛽’s fixed as well as Λ3,and recover straight away the decoupling limit of massive gravity on Minkowski found in (8.52),with a free and fully decoupled massless spin-2 field.

In the presence of a cosmological constant for both metrics (and thus a tadpole in this frame-work), we can also take the limit 𝑀𝑓 → ∞ and recover straight away the decoupling limit ofmassive gravity on (A)dS, as obtained in (8.66).

This illustrates the strength of this generic decoupling limit for bi-gravity (8.78). In principle wecould even go further and derive the decoupling limit of massive gravity on an arbitrary referencemetric as performed in [224]. To obtain a general reference metric we first need to add an externalsource for 𝑣𝜇𝜈 that generates a background for 𝑉𝜇𝜈 = 𝑀𝑓/𝑀Pl��𝜇𝜈 . The reference metric is thusexpressed in the local inertial frame as


𝑀𝑓𝑉𝜇𝜈 +

1

4𝑀2𝑓

𝑉𝜇𝛼𝑉𝛽𝜈𝜂𝛼𝛽 +

1

𝑀𝑓𝑣𝜇𝜈 +𝒪(𝑀−2

𝑓 ) (8.79)

= 𝜂𝜇𝜈 +1

𝑀Pl��𝜇𝜈 +

1

𝑀𝑓𝑣𝜇𝜈 +𝒪(𝑀Pl,𝑀𝑓 )

−2 . (8.80)

The fact that the metric 𝑓 looks like a perturbation away from Minkowski is related to the fact thatthe curvature needs to scale as 𝑚2 in the decoupling limit in order to avoid the issues previouslymentioned in the discussion of Section 8.2.3.

We can then perform the scaling limit 𝑀𝑓 → ∞, while keeping the 𝛽’s and the scale Λ3 =(𝑀Pl𝑚

2)1/3 fixed as well as the field 𝑣𝜇𝜈 and the fixed tensor ��𝜇𝜈 . The decoupling limit is thensimply given by

ℒ(U)Λ3

= ℒ(0)Λ3− 1

2��𝜇𝛽 [��]

(𝛿𝜈𝛽 +

Π𝜈𝛽Λ33

) 3∑𝑛=0

𝛽𝑛+1

Λ3(𝑛−1)3

𝑋(𝑛)𝜇𝜈 [Π] (8.81)

−1

4𝑣𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝑣𝛼𝛽 ,

where the helicity-2 field 𝑣 fully decouples from the rest of the massive gravity sector on the firstline which carries the other helicity-2 field as well as the helicity-1 and -0 modes. Notice that thegeneral metric �� has only an effect on the helicity-0 self-interactions, through the second term onthe first line of (8.81) (just as observed for the decoupling limit on AdS). These new interactions



Massive Gravity 89

are ghost-free and look like Galileons for conformally flat ��𝜇𝜈 = 𝜆𝜂𝜇𝜈 , with 𝜆 constant, but not ingeneral. In particular, the interactions found in (8.81) would not be the covariant Galileons foundin [166, 161, 157] (nor the ones found in [237]) for a generic metric.

9 Extensions of Ghost-free Massive Gravity

Massive gravity can be seen as a theory of a spin-2 field with the following free parameters inaddition to the standard parameters of GR (e.g., the cosmological constant, etc. . . ),

∙ Reference metric 𝑓𝑎𝑏,∙ Graviton mass 𝑚,∙ (𝑑− 2) dimensionless parameters 𝛼𝑛 (or the 𝛽’s) .

As natural extensions of massive gravity one can make any of these parameters dynamical. Asalready seen, the reference metric can be made dynamical leading to bi-gravity which in additionto massive spin-2 field carries a massless one as well.

Another natural extension is to promote the graviton mass 𝑚, or any of the free parameters 𝛼𝑛(or 𝛽𝑛) to a function of a new dynamical variable, say of an additional scalar field 𝜑. In principlethe mass 𝑚 and the parameters 𝛼’s can be thought as potentials for an arbitrary number of scalarfields 𝑚 = 𝑚(𝜓𝑗), 𝛼𝑛 = 𝛼𝑛(𝜓𝑗), and not necessarily the same fields for each one of them [320]. Solong as these functions are pure potentials and hide no kinetic terms for any new degree of freedom,the constraint analysis performed in Section 7 will go relatively unaffected, and the theory remainsfree from the BD ghost. This was shown explicitly for the mass-varying theory [319, 315] (where themass is promoted to a scalar function of a new single scalar field, 𝑚 = 𝑚(𝜑), while the parameters𝛼 remain constant23), as well as a general massive scalar-tensor theory [320], and for quasi-dilatonwhich allow for different couplings between the spin-2 and the scalar field, motivated by scaleinvariance. We review these models below in Sections 9.1 and 9.2.

Alternatively, rather than considering the parameters𝑚 and 𝛼 as arbitrary, one may set them tospecial values of special interest depending on the reference metric 𝑓𝜇𝜈 . Rather than an ‘extension’per se this is more special cases in the parameter space. The first obvious one is 𝑚 = 0 (forarbitrary reference metric and parameters 𝛼), for which one recovers the theory of GR (so long asthe spin-2 field couples to matter in a covariant way to start with). Alternatively, one may alsosit on the Higuchi bound, (see Section 8.3.6) with the parameters 𝑚2 = 2𝐻2, 𝛼3 = −1/3 and𝛼4 = 1/12 in four dimensions. This corresponds to the partially massless theory of gravity, whichat the moment is pathological in its simplest realization and will be reviewed below in Section 9.3.

The coupling massive gravity to a DBI Galileon [157] was considered in [237, 461, 261] leading toa generalized Galileon theory which maintains a Galileon symmetry on curved backgrounds. Thistheory was shown to be free of any Ostrogradsky ghost in [19] and the cosmology was recentlystudied in [315] and perturbations in [20].

Finally, as other extensions to massive gravity, one can also consider all the extensions applicableto GR. This includes the higher order Lovelock invariants in dimensions greater than four, as wellas promoting the Einstein–Hilbert kinetic term to a function 𝑓(𝑅), which is equivalent to gravitywith a scalar field. In the case of massive gravity this has been performed in [89] (see also [46, 354]),where the absence of BD ghost was proven via a constraint analysis, and the cosmology was explored(this was also discussed in Section 5.6 and see also Section 12.5). 𝑓(𝑅) extensions to bi-gravitywere also derived in [416, 415].

23 The non-renormalization theorem protects the parameters 𝛼’s and the mass from acquiring large quantumcorrections [140, 146] and it would be interesting to understand their implications in the case of a mass-varyinggravity.



90 Claudia de Rham

Trace-anomaly driven inflation in bi-gravity was also explored in Ref. [47]. Massless quantumeffects can be taking into account by including the trace anomaly 𝒯𝐴 given as [203]

𝒯𝐴 = 𝑐1(1

3𝑅2 − 2𝑅2

𝜇𝜈 +𝑅2𝜇𝜈𝛼𝛽 +

2

32𝑅) + 𝑐2(𝑅

2 − 4𝑅2𝜇𝜈 +𝑅2

𝜇𝜈𝛼𝛽) + 𝑐32𝑅 , (9.1)

where 𝑐1,2,3 are three constants depending on the field content (for instance the number of scalars,spinors, vectors, graviton etc.) Including this trace anomaly to the bi-gravity de Sitter-like solutionswere found which could represent a good model for anomaly-driven models of inflation.

9.1 Mass-varying

The idea behind mass-varying gravity is to promote the graviton mass to a potential for an externalscalar field 𝜓, 𝑚→ 𝑚(𝜓), which has its own dynamics [319], so that in four dimensions, the dRGTaction for massive gravity gets promoted to

ℒMass−Varying =𝑀2

Pl

2

∫d4𝑥√−𝑔

(𝑅+

𝑚2(𝜓)

2

4∑𝑛=0

𝛼𝑛ℒ𝑛[𝒦] (9.2)

−1

2𝑔𝜇𝜈𝜕𝜇𝜓𝜕𝜈𝜓 −𝑊 (𝜓)

),

and the tensors 𝒦 are given in (6.7). This could also be performed for bi-gravity, where we wouldsimply include the Einstein–Hilbert term for the metric 𝑓𝜇𝜈 . This formulation was then promotednot only to varying parameters 𝛼𝑛 → 𝛼𝑛(𝜓) but also to multiple fields 𝜓𝐴, with 𝐴 = 1, · · · ,𝒩in [320],

ℒGeneralized MG =𝑀2

Pl

2

∫d4𝑥√−𝑔[Ω(𝜓𝐴)𝑅+

1

2

4∑𝑛=0

𝛼𝑛(𝜓𝐴)ℒ𝑛[𝒦] (9.3)

−1

2𝑔𝜇𝜈𝜕𝜇𝜓𝐴𝜕𝜈𝜓

𝐴 −𝑊 (𝜓𝐴)].

The absence of BD ghost in these theories were performed in [319] and [320] in unitary gauge, inthe ADM language by means of a constraint analysis as formulated in Section 7.1. We recall thatin the absence of the scalar field 𝜓, the primary second-class (Hamiltonian) constraint is given by

𝒞0 = ℛ0(𝛾, 𝑝) +𝐷𝑖𝑗𝑛𝑗ℛ𝑖(𝛾, 𝑝) +𝑚2𝒰0(𝛾, 𝑛(𝛾, 𝑝)) ≈ 0 . (9.4)

In the case of a mass-varying theory of gravity, the entire argument remains the same, with thesimple addition of the scalar field contribution,

𝒞mass−varying0 = ℛ0(𝛾, 𝑝, 𝜓, 𝑝𝜓) +𝐷𝑖

𝑗𝑛𝑗ℛ𝑖(𝛾, 𝑝, 𝜓, 𝑝𝜓) +𝑚2(𝜓)𝒰0(𝛾, 𝑛(𝛾, 𝑝))

≈ 0 , (9.5)

where 𝑝𝜓 is the conjugate momentum associated with the scalar field 𝜓 and

ℛ0(𝛾, 𝑝, 𝜓, 𝑝𝜓) = ℛ0(𝛾, 𝑝) +1

2

√𝛾𝜕𝑖𝜓𝜕

𝑖𝜓 +1

2√𝛾𝑝2𝜓 (9.6)

ℛ𝑖(𝛾, 𝑝, 𝜓, 𝑝𝜓) = ℛ𝑖(𝛾, 𝑝) + 𝑝𝜓𝜕𝑖𝜓 . (9.7)

Then the time-evolution of this primary constraint leads to a secondary constraint similarly as inSection 7.1. The expression for this secondary constraint is the same as in (7.33) with a benign



Massive Gravity 91

new contribution from the scalar field [319]

𝒞2 = 𝒞2 +𝜕𝑚2(𝜓)

𝜕𝜓

[𝒰0𝜕𝑖𝜓(��𝑛𝑖 + �� 𝑖) +

��√𝛾𝒰1𝑝𝜓 + ��𝜕𝑖𝜓𝐷𝑖

𝑘𝑛𝑘

]≈ 0 . (9.8)

Then as in the normal fixed-mass case, the tertiary constraint is a constraint for the lapse and thesystem of constraint truncates leading to 5+1 physical degrees of freedom in four dimensions. Thesame logic goes through for generalized massive gravity as explained in [320].

One of the important aspects of a mass-varying theory of massive gravity is that it allows moreflexibility for the graviton mass. In the past the mass could have been much larger and could havelead to potential interesting features, be it for inflation (see for instance Refs. [315, 378] and [282]),the Hartle–Hawking no-boundary proposal [498, 439, 499], or to avoid the Higuchi bound [307], andyet be compatible with current bounds on the graviton mass. If the graviton mass is an effectivedescription from higher dimensions it is also quite natural to imagine that the graviton mass woulddepend on some moduli.

9.2 Quasi-dilaton

The Planck scale 𝑀Pl, or Newton constant explicitly breaks scale invariance, but one can easilyextend the theory of GR to a scale invariant one 𝑀Pl → 𝑀Pl𝑒

𝜆(𝑥) by including a dilaton scalarfield 𝜆 which naturally arises from string theory or from extra dimension compactification (see forinstance [122] and see Refs. [429, 120, 248] for the role of a dilaton scalar field on cosmology).

When dealing with multi-gravity, one can extend the notion of conformal transformation to theglobal rescaling of the coordinate system of one metric with respect to that of another metric. In thecase of massive gravity this amounts to considering the global rescaling of the reference coordinateswith respect to the physical one. As already seen, the reference metric can be promoted to a tensorwith respect to transformations of the physical metric coordinates, by introducing four Stuckelbergfields 𝜑𝑎, 𝑓𝜇𝜈 → 𝑓𝑎𝑏𝜕𝜇𝜑

𝑎𝜕𝜈𝜑𝑏. Thus the theory can be made invariant under global rescaling of

the reference metric if the reference metric is promoted to a function of the quasi-dilaton scalarfield 𝜎,

𝑓𝑎𝑏𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑏 → 𝑒2𝜎/𝑀Pl𝑓𝑎𝑏𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑏 . (9.9)

This is the idea behind the quasi-dilaton theory of massive gravity proposed in Ref. [119]. Thetheoretical consistency of this model was explored in [119] and is reviewed below. The Vainshteinmechanism and the cosmology were also explored in [119, 118] as well as in Refs. [288, 243, 127]and we review the cosmology in Section 12.5. As we shall see in that section, one of the interestsof quasi-dilaton massive gravity is the existence of spatially flat FLRW solutions, and particularlyof self-accelerating solutions. Nevertheless, such solutions have been shown to be strongly coupledwithin the region of interest [118], but an extension of that model was proposed in [127] and shownto be free from such issues.

Recently, the decoupling limit of the original quasi-dilaton model was derived in [239]. Interest-ingly, a new self-accelerating solution was found in this model which admits no instability and allthe modes are (sub)luminal for a given realistic set of parameters. The extension of this solutionto the full theory (beyond the decoupling limit) should provide for a consistent self-acceleratingsolution which is guaranteed to be stable (or with a harmless instability time scale of the order ofthe age of the Universe at least).

9.2.1 Theory

As already mentioned, the idea behind quasi-dilaton massive gravity (QMG) is to extend massivegravity to a theory which admits a new global symmetry. This is possible via the introduction of



92 Claudia de Rham

a quasi-dilaton scalar field 𝜎(𝑥). The action for QMG is thus given by

𝑆QMG =𝑀2

Pl

2

∫d4𝑥√−𝑔[𝑅− 𝜔

2𝑀2Pl

(𝜕𝜎)2 +𝑚2

2

4∑𝑛=0

𝛼𝑛ℒ𝑛[��[𝑔, 𝜂]]]

(9.10)

+

∫d4𝑥√−𝑔ℒmatter(𝑔, 𝜓) ,

where 𝜓 represent the matter fields, 𝑔 is the dynamical metric, and unless specified otherwise allindices are raised and lowered with respect to 𝑔, and 𝑅 represents the scalar curvature with respectto 𝑔. The Lagrangians ℒ𝑛 were expressed in (6.9 – 6.13) or (6.14 – 6.18) and the tensor �� is givenin terms of the Stuckelberg fields as

��𝜇𝜈 [𝑔, 𝜂] = 𝛿𝜇𝜈 − 𝑒𝜎/𝑀Pl

√𝑔𝜇𝛼𝜕𝛼𝜑𝑎𝜕𝜈𝜑𝑏𝜂𝑎𝑏 . (9.11)

In the case of the QMG presented in [119], there is no cosmological constant nor tadpole (𝛼0 = 𝛼1 =0) and 𝛼2 = 1. This is a very special case of the generalized theory of massive gravity presentedin [320], and the proof for the absence of BD ghost thus goes through in the same way. Here againthe presence of the scalar field brings only minor modifications to the Hamiltonian analysis in theADM language as presented in Section 9.1, and so we do not reproduce the proof here. We simplynote that the theory propagates six degrees of freedom in four dimensions and is manifestly free ofany ghost on flat space time provided that 𝜔 > 1/6. The key ingredient compared to mass-varyinggravity or generalized massive gravity is the presence of a global rescaling symmetry which is botha space-time and internal transformation [119],

𝑥𝜇 → 𝑒𝜉𝑥𝜇, 𝑔𝜇𝜈 → 𝑒−2𝜉𝑔𝜇𝜈 , 𝜎 → 𝜎 −𝑀Pl𝜉, and 𝜑𝑎 → 𝑒𝜉𝜑𝑎 . (9.12)

Notice that the matter action d4𝑥√−𝑔ℒ(𝑔, 𝜓) breaks this symmetry, reason why it is called a

‘quasi -dilaton’.An interesting feature of QMG is the fact that the decoupling limit leads to a bi-Galileon theory,

one Galileon being the helicity-0 mode presented in Section 8.3, and the other Galileon being thequasi-dilaton 𝜎. Just as in massive gravity, there are no irrelevant operators arising at energy scalebelow Λ3, and at that scale the theory is given by

ℒ(QMG)Λ3

= ℒ(0)Λ3− 𝜔

2(𝜕𝜎)2 +

1

2𝜎

4∑𝑛=1

(4− 𝑛)𝛼𝑛 − (𝑛+ 1)𝛼𝑛+1

Λ3(𝑛−1)3

ℒ𝑛[Π] , (9.13)

where the decoupling limit Lagrangian ℒ(0)Λ3

in the absence of the quasi-dilaton is given in (8.52) andwe recall that 𝛼2 = 1, 𝛼1 = 0, Π𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋 and the Lagrangians ℒ𝑛 are expressed in (6.10) – (6.13)or (6.15) – (6.18). We see emerging a bi-Galileon theory for 𝜋 and 𝜎, and thus the decoupling limitis manifestly ghost-free. We could then apply a similar argument as in Section 7.2.4 to infer theabsence of BD ghost for the full theory based on this decoupling limit. Up to integration by parts,the Lagrangian (9.13) is invariant under both independent Galilean transformation 𝜋 → 𝜋+𝑐+𝑣𝜇𝑥

𝜇

and 𝜎 → 𝜎 + 𝑐+ 𝑣𝜇𝑥𝜇.

One of the relevance of this decoupling limit is that it makes the study of the Vainshtein mech-anism more explicit. As we shall see in what follows (see Section 10.1), the Galileon interactionsare crucial for the Vainshtein mechanism to work.

Note that in (9.13), the interactions with the quasi-dilaton come in the combination ((4 −𝑛)𝛼𝑛 − (𝑛+ 1)𝛼𝑛+1), while in ℒ(0)

Λ3, the interactions between the helicity-0 and -2 modes come in

the combination ((4− 𝑛)𝛼𝑛 + (𝑛+ 1)𝛼𝑛+1). This implies that in massive gravity, the interactionsbetween the helicity-2 and -0 mode disappear in the special case where 𝛼𝑛 = −(𝑛+1)/(4−𝑛)𝛼𝑛+1



Massive Gravity 93

(this corresponds to the minimal model), and the Vainshtein mechanism is no longer active forspherically symmetric sources (see Refs. [99, 56, 58, 57, 435]). In the case of QMG, the interactionswith the quasi-dilaton survive in that specific case 𝛼3 = −4𝛼4, and a Vainshtein mechanismcould still be feasible, although one might still need to consider non-asymptotically Minkowskiconfigurations.

The cosmology of QMD was first discussed in [119] where the existence of self-acceleratingsolutions was pointed out. This will be reviewed in the section on cosmology, see Section 12.5. Wenow turn to the extended version of QMG recently proposed in Ref. [127].

9.2.2 Extended quasi-dilaton

Keeping the same philosophy as the quasi-dilaton in mind, a simple but yet powerful extension wasproposed in Ref. [127] and then further extended in [126], leading to interesting phenomenologyand stable self-accelerating solutions. The phenomenology of this model was then further exploredin [45]. The stability of the extended quasi-dilaton theory of massive gravity was explored in [353]and was proven to be ghost-free in [406].

The key ingredient behind the extended quasi-dilaton theory of massive gravity (EMG) is tonotice that two most important properties of QMG namely the absence of BD ghost and theexistence of a global scaling symmetry are preserved if the covariantized reference metric is furthergeneralized to include a disformal contribution of the form 𝜕𝜇𝜎𝜕𝜈𝜎 (such a contribution to thereference metric can arise naturally from the brane-bending mode in higher dimensional braneworldmodels, see for instance [157]).

The action for EMG then takes the same form as in (9.10) with the tensor �� promoted to

�� → �� = I− 𝑒𝜎/𝑀Pl

√𝑔−1𝑓 , (9.14)

with the tensor 𝑓𝜇𝜈 defined as

𝑓𝜇𝜈 = 𝜕𝜇𝜑𝑎𝜕𝜈𝜑

𝑏𝜂𝑎𝑏 −𝛼𝜎

𝑀PlΛ33

𝑒−2𝜎/𝑀Pl𝜕𝜇𝜎𝜕𝜈𝜎 , (9.15)

where 𝛼𝜎 is a new coupling dimensionless constant (as mentioned in [127], this coupling constantis expected to enjoy a non-renormalization theorem in the decoupling limit, and thus to receivequantum corrections which are always suppressed by at least 𝑚2/Λ2

3). Furthermore, this actioncan be generalized further by

∙ Considering different coupling constants for the ��’s entering in ℒ2[��], ℒ3[��] and ℒ4[��].∙ One can also introduce what would be a cosmological constant for the metric 𝑓 , namely anew term of the form

√−𝑓𝑒4𝜎/𝑀Pl .

∙ General shift-symmetric Horndeski Lagrangians for the quasi-dilaton.

With these further generalizations, one can obtain self-accelerating solutions similarly as in theoriginal QMG. For these self-accelerating solutions, the coupling constant 𝛼𝜎 does not enter thebackground equations of motion but plays a crucial role for the stability of the scalar perturbationson top of these solutions. This is one of the benefits of this extended quasi-dilaton theory of massivegravity.



94 Claudia de Rham

9.3 Partially massless

9.3.1 Motivations behind PM gravity

The multiple proofs for the absence of BD ghost presented in Section 7 ensures that the ghost-free theory of massive gravity, (or dRGT) does not propagate more than five physical degrees offreedom in the graviton. For a generic finite mass 𝑚 the theory propagates exactly five degreesof freedom as can be shown from a linear analysis about a generic background. Yet, one can askwhether there exists special points in parameter space where some of degrees of freedom decouple.General relativity, for which 𝑚 = 0 (and the other parameters 𝛼𝑛 are finite) is one such example.In the massless limit of massive gravity the two helicity-1 modes and the helicity-0 mode decouplefrom the helicity-2 mode and we thus recover the theory of a massless spin-2 field correspondingto GR, and three decoupled degrees of freedom. The decoupling of the helicity-0 mode occurs viathe Vainshtein mechanism24 as we shall see in Section 10.1.

As seen in Section 8.3.6, when considering massive gravity on de Sitter as a reference metric,if the graviton mass is precisely 𝑚2 = 2𝐻2, the helicity-0 mode disappears linearly as can be seenfrom the linearized Lagrangian (8.62). The same occurs in any dimension when the graviton massis tied to the de Sitter curvature by the relation 𝑚2 = (𝑑−2)𝐻2. This special case is another pointin parameter space where the helicity-0 mode could be decoupled, corresponding to a partiallymassless (PM) theory of gravity as first pointed out by Deser and Waldron [190, 189, 188], (seealso [500] for partially massless higher spin, and [450] for related studies).

The absence of helicity-0 mode at the linearized level in PM is tied to the existence of anew scalar gauge symmetry at the linearized level when 𝑚2 = 2𝐻2 (or (𝑑 − 2)𝐻2 in arbitrarydimensions), which is responsible for making the helicity-0 mode unphysical. Indeed the ac-tion (8.62) is invariant under a special combination of a linearized diff and a conformal trans-formation [190, 189, 188],

ℎ𝜇𝜈 → ℎ𝜇𝜈 +∇𝜇∇𝜈𝜉 − (𝑑− 2)𝐻2𝜉𝛾𝜇𝜈 . (9.16)

If a non-linear completion of PM gravity exist, then there must exist a non-linear completion of thissymmetry which eliminates the helicity-0 mode to all orders. The existence of such a symmetrywould lead to several outstanding features:

∙ It would protect the structure of the potential.∙ In the PM limit of massive gravity, the helicity-0 mode fully decouples from the helicity-2mode and hence from external matter. As a consequence, there is no Vainshtein mechanismthat decouples the helicity-0 mode in the PM limit of massive gravity unlike in the masslesslimit. Rather, the helicity-0 mode simply decouples without invoking any strong couplingeffects and the theoretical and observational luggage that goes with it.∙ Last but not least, in PM gravity the symmetry underlying the theory is not diffeomorphisminvariance but rather the one pointed out in (9.16). This means that in PM gravity, anarbitrary cosmological constant does not satisfy the symmetry (unlike in GR). Rather, thevalue of the cosmological constant is fixed by the gauge symmetry and is proportional tothe graviton mass. As we shall see in Section 10.3 the graviton mass does not receive largequantum corrections (it is technically natural to set to small values). So, if a PM theory ofgravity existed it would have the potential to tackle the cosmological constant problem.

Crucially, breaking of covariance implies that matter is no longer covariantly conserved. Insteadthe failure of energy conservation is proportional to the graviton mass,

∇𝜇∇𝜈𝑇𝜇𝜈 = − 𝑚2

𝑑− 2𝑇 , (9.17)

24 Note that the Vainshtein mechanism does not occur for all parameters of the theory. In that case the masslesslimit does not reproduce GR.



Massive Gravity 95

which in practise is extremely small.It is worth emphasizing that if a PM theory of gravity existed, it would be distinct from the

minimal model of massive gravity where the non-linear interactions between the helicity-0 and -2modes vanish in the decoupling limit but the helicity-0 mode is still fully present. PM gravity isalso distinct from some specific branches of solutions found in cosmology (see Section 12) on top ofwhich the helicity-0 mode disappears. If a PM theory of gravity exists the helicity-0 mode wouldbe fully absent of the whole theory and not only for some specific branches of solutions.

9.3.2 The search for a PM theory of gravity

A candidate for PM gravity:

The previous considerations represent some strong motivations for finding a fully fledged theoryof PM gravity (i.e., beyond the linearized theory) and there has been many studies to find a non-linear realization of the PM symmetry. So far all these studies have in common to keep the kineticterm for gravity unchanged (i.e., keeping the standard Einstein–Hilbert action, with a potentialgeneralization to the Lovelock invariants [298]).

Under this assumption, it was shown in [501, 330], that while the linear level theory admits asymmetry in any dimensions, at the cubic level the PM symmetry only exists in 𝑑 = 4 spacetimedimensions, which could make the theory even more attractive. It was also pointed out in [191]that in four dimensions the theory is conformally invariant. Interestingly, the restriction to fourdimensions can be lifted in bi-gravity by including the Lovelock invariants [298].

From the analysis in Section 8.3.6 (see Ref. [154]) one can see that the helicity-0 mode entirelydisappears from the decoupling limit of ghost-free massive gravity, if one ignores the vectors andsets the parameters of the theory to 𝑚2 = 2𝐻2, 𝛼3 = −1 and 𝛼4 = 1/4 in four dimensions.The ghost-free theory of massive gravity with these parameters is thus a natural candidate forthe PM theory of gravity. Following this analysis, it was also shown that bi-gravity with thesame parameters for the interactions between the two metrics satisfies similar properties [301].Furthermore, it was also shown in [147] that the potential has to follow the same structure as thatof ghost-free massive gravity to have a chance of being an acceptable candidate for PM gravity.In bi-gravity the same parameters as for massive gravity were considered as also being the naturalcandidate [301], in addition of course to other parameters that vanish in the massive gravity limit(to make a fair comparison once needs to take the massive gravity limit of bi-gravity with care aswas shown in [301]).

Re-appearance of the Helicity-0 mode:

Unfortunately, when analyzing the interactions with the vector fields, it is clear from the decouplinglimit (8.52) that the helicity-0 mode reappears non-linearly through their couplings with the vectorfields. These never cancel, not even in four dimensions and for no parameters of theory. So ratherthan being free from the helicity-0 mode, massive gravity with 𝑚2 = (𝑑 − 2)𝐻2 has an infinitelystrongly coupled helicity-0 mode and is thus a sick theory. The absence of the helicity-0 mode issimple artefact of the linear theory.

As a result we can thus deduce that there is no theory of PM gravity. This result is consistentwith many independent studies performed in the literature (see Refs. [185, 147, 181, 194]).

Relaxing the assumptions:

∙ One assumption behind this result is the form of the kinetic term for the helicity-2 mode,which is kept to the be Einstein–Hilbert term as in GR. A few studies have considereda generalization of that kinetic term to diffeomorphism-breaking ones [231, 310] however



96 Claudia de Rham

further analysis [339, 153] have shown that such interactions always lead to ghosts non-perturbatively. See Section 5.6 for further details.

∙ Another potential way out is to consider the embedding of PM within bi-gravity or multi-gravity. Since bi-gravity is massive gravity and a decoupled massless spin-2 field in somelimit it is unclear how bi-gravity could evade the results obtained in massive gravity butthis approach has been explored in [301, 298, 299, 184]. A perturbative relation between bi-gravity and conformal gravity was derived at the level of the equations of motion in Ref. [299](unlike claimed in [184]).

∙ The other assumptions are locality and Lorentz-invariance. It is well known that Lorentz-breaking theories of massive gravity can excite fewer than five degrees of freedom. Thisavenue is explored in Section 14.

To summarize there is to date no known non-linear PM symmetry which could project out thehelicity-0 mode of the graviton while keeping the helicity-2 mode massive in a local and Lorentzinvariant way.



Massive Gravity 97

10 Massive Gravity Field Theory

10.1 Vainshtein mechanism

As seen earlier, in four dimensions a massless spin-2 field has five degrees of freedom, and there isno special PM case of gravity where the helicity-0 mode is unphysical while the graviton remainsmassive (or at least there is to date no known such theory). The helicity-0 mode couples to matteralready at the linear level and this additional coupling leads to a extra force which is at the originof the vDVZ discontinuity see in Section 2.2.3. In this section, we shall see how the non-linearitiesof the helicity-0 mode is responsible for a Vainshtein mechanism that screens the effect of this fieldin the vicinity of matter.

Since the Vainshtein mechanism relies strongly on non-linearities, this makes explicit solutionsvery hard to find. In most of the cases where the Vainshtein mechanism has been shown towork successfully, one assumes a static and spherically symmetric background source. Already inthat case the existence of consistent solutions which extrapolate from a well-behaved asymptoticbehavior at infinity to a screened solution close to the source are difficult to obtain numerically [121]and were only recently unveiled [37, 39] in the case of non-linear Fierz–Pauli gravity.

This review on massive gravity cannot do justice to all the ongoing work dedicated to the studyof the Vainshtein mechanism (also sometimes called ‘kinetic chameleon’ as it relies on the kineticinteractions for the helicity-0 mode). In what follows, we will give the general idea behind theVainshtein mechanism starting from the decoupling limit of massive gravity and then show explicitsolutions in the decoupling limit for static and spherically symmetric sources. Such an analysis isrelevant for observational tests in the solar system as well as for other astrophysical tests (such asbinary pulsar timing), which we shall explore in Section 11. We refer to the following review onthe Vainshtein mechanism for further details, [35] as well as to the following work [160, 38, 99, 332,36, 244, 40, 338, 321, 440, 316, 53, 376, 366, 407]. Recently, it was also shown that the Vainshteinmechanism works for bi-gravity, see Ref. [34].

We focus the rest of this section to the case of four space-time dimensions, although many ofthe results presented in what follows are well understood in arbitrary dimensions.

10.1.1 Effective coupling to matter

As already mentioned, the key ingredient behind the Vainshtein mechanism is the importance ofinteractions for the helicity-0 mode which we denote as 𝜋. From the decoupling limit analysisperformed for massive gravity (see (8.52)) and bi-gravity (see (8.78)), we see that in some limitthe helicity-0 mode 𝜋 behaves as a scalar field, which enjoys a special global symmetry

𝜋 → 𝜋 + 𝑐+ 𝑣𝜇𝑥𝜇 , (10.1)

and yet only carries two derivatives at the level of the equations of motion, (which as we have seenis another way to see the absence of BD ghost).

These types of interactions are very similar to the Galileon-type of interactions introduced byNicolis, Rattazzi and Trincherini in Ref. [412] as a generalization of the decoupling limit of DGP.For simplicity we shall focus most of the discussion on the Vainshtein mechanism with Galileonsas a special example, and then mention in Section 10.1.3 peculiarities that arise in the special caseof massive gravity (see for instance Refs. [58, 57]).

We thus start with a cubic Galileon theory

ℒ = −1

2(𝜕𝜋)2 − 1

Λ3(𝜕𝜋)22𝜋 +

1

𝑀Pl𝜋𝑇, , (10.2)

where 𝑇 = 𝑇𝜇𝜇 is the trace of the stress-energy tensor of external sources, and Λ is the strong

coupling scale of the theory. As seen earlier, in the case of massive gravity, Λ = Λ3 = (𝑚2𝑀Pl)1/3.



98 Claudia de Rham

This is actually precisely the way the helicity-0 mode enters in the decoupling limit of DGP [389]as seen in Section 4.2. It is in that very context that the Vainshtein mechanism was first shown towork explicitly [165].

The essence of the Vainshtein mechanism is that close to a source, the Galileon interactionsdominate over the linear piece. We make use of this fact by splitting the source into a backgroundcontribution 𝑇0 and a perturbation 𝛿𝑇 . The background source 𝑇0 leads to a background profile𝜋0 for the field, and the response to the fluctuation 𝛿𝑇 on top of this background is given by 𝜑, sothat the total field is expressed as

𝜋 = 𝜋0 + 𝜑 . (10.3)

For a sufficiently large source (or as we shall see below if 𝑇0 represents a static point-like source,then sufficiently close to the source), the non-linearities dominate and symbolically 𝜕2𝜋0 ≫ Λ3.

We now follow the perturbations in the action (10.2) and notice that the background configu-ration 𝜋0 leads to a modified effective metric for the perturbations,

ℒ(2) = −1

2𝑍𝜇𝜈(𝜋0)𝜕𝜇𝜑𝜕𝜈𝜑+

1

𝑀Pl𝜑𝛿𝑇 , (10.4)

up to second order in perturbations, with the new effective metric 𝑍𝜇𝜈

𝑍𝜇𝜈 = 𝜂𝜇𝜈 +2

Λ3𝑋(1)𝜇𝜈(Π0) , (10.5)

where the tensor 𝑋(1) is the same as that defined for massive gravity in (8.29) or in (8.34), so

symbolically 𝑍 is of the form 𝑍 ∼ 1 + 𝜕2𝜋0

Λ3 . One can generalize the initial action (10.2) toarbitrary set of Galileon interactions

ℒ = 𝜋4∑

𝑛=1

𝑐𝑛+1

Λ3(𝑛−1)ℒ𝑛[Π] , (10.6)

with again Π𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋 and where the scalars ℒ𝑛 have been defined in (6.10) – (6.13). The effectivemetric would then be of the form

𝑍𝜇𝜈(𝜋0) =4∑

𝑛=1

𝑛(𝑛+ 1)𝑐𝑛Λ3(𝑛−1)

𝑋(𝑛−1)𝜇𝜈(Π0) , (10.7)

where all the tensors 𝑋(𝑛)𝜇𝜈 are defined in (8.28) – (8.32). Notice that 𝜕𝜇𝑍

𝜇𝜈 = 0 identically. Forsufficiently large sources, the components of 𝑍 are large, symbolically, 𝑍 ∼

(𝜕2𝜋0/Λ

3)𝑛 ≫ 1 for

𝑛 ≥ 1.Canonically normalizing the fluctuations in (10.4), we have symbolically,

𝜑 =√𝑍𝜑 , (10.8)

assuming 𝑍𝜇𝜈 ∼ 𝑍𝜂𝜇𝜈 , which is not generally the case. Nevertheless, this symbolic scaling issufficient to get the essence of the idea. For a more explicit canonical normalization in specificconfigurations see Ref. [412]. As nicely explained in that reference, if 𝑍𝜇𝜈 is conformally flat, one

should not only scale the field 𝜑 → 𝜑 but also the space-like coordinates 𝑥 → �� so at to obtain astandard canonically normalized field in the new system,

∫d4�� − 1

2 (𝜕��𝜑)2. For now we stick to

the simple normalization (10.8) as it is sufficient to see the essence of the Vainshtein mechanism.

In terms of the canonically normalized field 𝜑, the perturbed action (10.4) is then

ℒ(2) = −1

2(𝜕𝜑)2 +

1

𝑀Pl

√𝑍(𝜋0)

𝜑𝛿𝑇 , (10.9)



Massive Gravity 99

which means that the coupling of the fluctuations to matter is medium dependent and can ariseat a scale very different from the Planck scale. In particular, for a large background configuration,𝜕2𝜋0 ≫ Λ3 and 𝑍(𝜋0)≫ 1, so the effective coupling scale to external matter is

𝑀eff =𝑀Pl

√𝑍 ≫𝑀Pl , (10.10)

and the coupling to matter is thus very suppressed. In massive gravity Λ is related to the gravitonmass, Λ ∼ 𝑚2/3, and so the effective coupling scale 𝑀eff → ∞ as 𝑚 → 0, which shows how thehelicity-0 mode characterized by 𝜋 decouples in the massless limit.

We now first review how the Vainshtein mechanism works more explicitly in a static andspherically symmetric configuration before applying it to other systems. Note that the Vainshteinmechanism relies on irrelevant operators. In a standard EFT this cannot be performed withoutgoing beyond the regime of validity of the EFT. In the context of Galileons and other very specificderivative theories, one can reorganize the EFT so that the operators considered can be large andyet remain within the regime of validity of the reorganized EFT. This will be discussed in moredepth in what follows.

10.1.2 Static and spherically symmetric configurations in Galileons

Suppression of the force

We now consider a point like source

𝑇0 = −𝑀𝛿(3)(𝑟) = −𝑀 𝛿(𝑟)

4𝜋𝑟2, (10.11)

where 𝑀 is the mass of the source localized at 𝑟 = 0. Since the source is static and sphericallysymmetric, we can focus on configurations which respect the same symmetry, 𝜋0 = 𝜋0(𝑟). Thebackground configuration for the field 𝜋0(𝑟) in the case of the cubic Galileon (10.2) satisfies theequation of motion [411]

1

𝑟2𝜕𝑟

[𝑟3

(𝜋′0(𝑟)

𝑟+

1

Λ3

(𝜋′0(𝑟)

𝑟

)2)]

=𝑀

4𝜋𝑀Pl

𝛿(𝑟)

𝑟2, (10.12)

and so integrating both sides of the equation, we obtain an algebraic equation for 𝜋′0(𝑟),

𝜋′0(𝑟)

𝑟+

1

Λ3

(𝜋′0(𝑟)

𝑟

)2

=𝑀

𝑀Pl

1

4𝜋𝑟3. (10.13)

We can define the Vainshtein or strong coupling radius 𝑟* as

𝑟* =1

Λ

(𝑀

4𝜋𝑀Pl

)1/3

, (10.14)

so that at large distances compared to that Vainshtein radius the linear term in (10.12) dominateswhile the interactions dominate at distances shorter than 𝑟*,

for 𝑟 ≫ 𝑟*, 𝜋′0(𝑟) ∼

𝑀

4𝜋𝑀Pl

1

𝑟2

for 𝑟 ≪ 𝑟*, 𝜋′0(𝑟) ∼

𝑀

4𝜋𝑀Pl

1

𝑟3/2* 𝑟1/2

. (10.15)

So, at large distances 𝑟 ≫ 𝑟*, one recovers a Newton square law for the force mediated by 𝜋, andthat fields mediates a force which is just a strong as standard gravity (i.e., as the force mediated by



100 Claudia de Rham

the usual helicity-2 modes of the graviton). On shorter distances scales, i.e., close to the localizedsource, the force mediated by the new field 𝜋 is much smaller than the standard gravitational one,

𝐹(𝜋)𝑟≪𝑟*

𝐹Newt∼(𝑟

𝑟*

)3/2

≪ 1 for 𝑟 ≪ 𝑟⋆ . (10.16)

In the case of the quartic Galileon (which typically arises in massive gravity), the force is evensuppressed and goes as

𝐹(quartic 𝜋)𝑟≪𝑟*

𝐹Newt∼(𝑟

𝑟*

)2

≪ 1 for 𝑟 ≪ 𝑟⋆ . (10.17)

For a graviton mass of the order of the Hubble parameter today, i.e., Λ ∼ (1000 km)−1, then takinginto account the mass of the Sun, the force at the position of the Earth is suppressed by 12 ordersof magnitude compared to standard Newtonian force in the case of the cubic Galileon and by 16orders of magnitude in the quartic Galileon. This means that the extra force mediated by 𝜋 isutterly negligible compared to the standard force of gravity and deviations to GR are extremelysmall.

Considering the Earth-Moon system, the force mediated by 𝜋 at the surface of the Moonis suppressed by 13 orders of magnitude compared to the Newtonian one in the cubic Galileon.While small, this is still not far off from the possible detectability from the lunar laser ranging spaceexperiment [488], as will be discussed further in what follows. Note that in the quartic Galileon,that force is suppressed instead by 17 orders of magnitude and is there again very negligible.

When applying this naive estimate (10.16) to the Hulse–Taylor system for instance, we wouldinfer a suppression of 15 orders of magnitude compared to the standard GR results. As we shall seein what follows this estimate breaks down when the time evolution is not negligible. These pointswill be discussed in the phenomenology Section 11, but before considering these aspects we reviewin what follows different aspects of massive gravity from a field theory perspective, emphasizingthe regime of validity of the theory as well as the quantum corrections that arise in such a theoryand the emergence of superluminal propagation.

Perturbations

We now consider perturbations riding on top of this background configuration for the Galileonfield, 𝜋 = 𝜋0(𝑟)+𝜑(𝑥

𝜇). As already derived in Section 10.1.1, the perturbations 𝜑 see the effectivespace-dependent metric 𝑍𝜇𝜈 given in (10.7). Focusing on the cubic Galileon for concreteness, thebackground solution for 𝜋0 is given by (10.13). In that case the effective metric is

𝑍𝜇𝜈 = 𝜂𝜇𝜈 +4

Λ3(2𝜋0𝜂

𝜇𝜈 − 𝜕𝜇𝜕𝜈𝜋0) (10.18)

𝑍𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −(1 +

4

Λ3

(2𝜋′

0(𝑟)

𝑟+ 𝜋′′

0 (𝑟)

))d𝑡2 (10.19)

+

(1 +

8𝜋′0(𝑟)

𝑟Λ3

)d𝑟2 +

(1 +

4

Λ3

(𝜋′0(𝑟)

𝑟+ 𝜋′′

0 (𝑟)

))𝑟2 dΩ2

2 ,

so that close to the source, for 𝑟 ≪ 𝑟*,

𝑍𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = 6(𝑟*𝑟

)1/2(− d𝑡2 +

4

3d𝑟2 +

1

3𝑟2 dΩ2

2

)+𝒪(𝑟*/𝑟)0 . (10.20)

A few comments are in order:



Massive Gravity 101

∙ First, we recover 𝑍 ∼√𝑟*/𝑟 ≫ 1 for 𝑟 ≪ 𝑟*, which is responsible for the redressing of

the strong coupling scale as we shall see in (10.24). On the no-trivial background the newstrong coupling scale is Λ* ∼

√𝑍Λ ≫ Λ for 𝑟 ≪ 𝑟*. Similarly, on top of this background

the coupling to external matter no longer occurs at the Planck scale but rather at the scale√𝑍𝑀Pl ∼ 107𝑀Pl.

∙ Second, we see that within the regime of validity of the classical calculation, the modespropagating along the radial direction do so with a superluminal phase and group velocity𝑐2𝑟 = 4/3 > 1 and the modes propagating in the orthoradial direction do so with a subluminalphase and group velocity 𝑐2Ω = 1/3. This result occurs in any Galileon and multi-Galileontheory which exhibits the Vainshtein mechanism [412, 129, 246]. The subluminal velocity isnot of great concern, not even for Cerenkov radiation since the coupling to other fields is somuch suppressed, but the superluminal velocity has been source of many questions [1]. It isdefinitely one of the biggest issues arising in these kinds of theories see Section 10.6.

Before discussing the biggest concerns of the theory, namely the superluminalities and the lowstrong-coupling scale, we briefly present some subtleties that arise when considering static andspherically symmetric solutions in massive gravity as opposed to a generic Galileon theory.

10.1.3 Static and spherically symmetric configurations in massive gravity

The Vainshtein mechanism was discussed directly in the context of massive gravity (rather thanthe Galileon larger family) in Refs. [363, 365, 99, 440] and more recently in [58, 455, 57]. See alsoRefs. [478, 105, 61, 413, 277, 160, 38, 37, 39] for other spherically symmetric solutions in massivegravity.

While the decoupling limit of massive gravity resembles that of a Galileon, it presents a fewparticularities which affects the precise realization of the Vainshtein mechanism:

∙ First if the parameters of the ghost-free theory of massive gravity are such that 𝛼3+4𝛼4 = 0,

there is a mixing ℎ𝜇𝜈𝑋(3)𝜇𝜈 between the helicity-0 and -2 modes of the graviton that cannot be

removed by a local field redefinition (unless we work in an special types of backgrounds). Theeffects of this coupling were explored in [99, 57] and it was shown that the theory does notexhibit any stable static and spherically symmetric configuration in presence of a localizedpoint-like matter source. So in order to be phenomenologically viable, the theory of massivegravity needs to be tuned with 𝛼3+4𝛼4 = 0. Since these parameters do not get renormalizedthis is a tuning and not a fine-tuning.

∙ When 𝛼3+4𝛼4 = 0 and the previous mixing ℎ𝜇𝜈𝑋(3)𝜇𝜈 is absent, the decoupling limit of massive

gravity resembles a specific quartic Galileon, where the coefficient of the cubic Galileon isrelated to quartic coefficient (and if one vanishes so does the other one),

ℒHelicity−0 = −3

4(𝜕𝜋)2 +

3𝛼

4Λ33

ℒ(3)(Gal)[𝜋]−

1

4

(𝛼

Λ33

)2

ℒ(4)(Gal)[𝜋] (10.21)

+1

𝑀Pl

(𝜋𝑇 +

𝛼

Λ33

𝜕𝜇𝜋𝜕𝜈𝜋𝑇𝜇𝜈

),

where we have set 𝛼2 = 1 and the Galileon Lagrangians ℒ(3,4)(Gal)[𝜋] are given in (8.44) and

(8.45). Note that in this decoupling limit the graviton mass always enters in the combination𝛼/Λ3

3, with 𝛼 = −(1 + 3/2𝛼3). As a result this decoupling limit can never be used todirectly probe the graviton mass itself but rather of the combination 𝛼/Λ3

3 [57]. Beyond thedecoupling limit however the theory breaks the degeneracy between 𝛼 and 𝑚.



102 Claudia de Rham

Not only is the cubic Galileon always present when the quartic Galileon is there, but onecannot prevent the new coupling to matter 𝜕𝜇𝜋𝜕𝜈𝜕𝜋𝑇

𝜇𝜈 which is typically absent in otherGalileon theories.

The effect of the coupling 𝜕𝜇𝜋𝜕𝜈𝜋𝑇𝜇𝜈 was explored in [58]. First it was shown that this coupling

contributes to the definition of the kinetic term of 𝜋 and can lead to a ghost unless 𝛼 > 0 so thisrestricts further the allowed region of parameter space for massive gravity. Furthermore, even when𝛼 > 0, none of the static spherically symmetric solutions which asymptote to 𝜋 → 0 at infinity(asymptotically flat solutions) extrapolate to a Vainshtein solution close to the source. Insteadthe Vainshtein solution near the source extrapolate to cosmological solutions at infinity which isindependent of the source

𝜋0(𝑟)→3 +√3

4

Λ33

𝛼𝑟2 for 𝑟 ≫ 𝑟* (10.22)

𝜋0(𝑟)→(Λ33

𝛼

)2/3(𝑀

4𝜋𝑀Pl

)1/3

for 𝑟 ≪ 𝑟* . (10.23)

If 𝜋 was a scalar field in its own right such an asymptotic condition would not be acceptable.However, in massive gravity 𝜋 is the helicity-0 mode of the gravity and its effect always entersfrom the Stuckelberg combination 𝜕𝜇𝜕𝜈𝜋, which goes to a constant at infinity. Furthermore, thisresult is only derived in the decoupling limit, but in the fully fledged theory of massive gravity, thegraviton mass kicks in at the distance scale ℓ ∼ 𝑚−1 and suppresses any effect at these scales.

Interestingly, when performing the perturbation analysis on this solution, the modes along alldirections are subluminal, unlike what was found for the Galileon in (10.20). It is yet unclearwhether this is an accident to this specific solution or if this is something generic in consistentsolutions of massive gravity.

10.2 Validity of the EFT

The Vainshtein mechanism presented previously relies crucially on interactions which are importantat a low energy scale Λ ≪ 𝑀Pl. These interactions are operators of dimension larger than four,for instance the cubic Galileon (𝜕𝜋)22𝜋 is a dimension-7 operator and the quartic Galileon is adimension-10 operator. The same can be seen directly within massive gravity. In the decoupling

limit (8.38), the terms ℎ𝜇𝜈𝑋(2,3)𝜇𝜈 are respectively dimension-7 and-10 operators. These operators

are thus irrelevant from a traditional EFT viewpoint and the theory is hence not renormalizable.This comes as no surprise, since gravity itself is not renormalizable and there is thus no reason to

expect massive gravity nor its decoupling limit to be renormalizable. However, for the Vainshteinmechanism to be successful in massive gravity, we are required to work within a regime wherethese operators dominate over the marginal ones (i.e., over the standard kinetic term (𝜕𝜋)2 in thestrongly coupled region where 𝜕2𝜋 ≫ Λ3). It is, therefore, natural to wonder whether or not onecan ever use the effective field description within the strong coupling region without going outsidethe regime of validity of the theory.

The answer to this question relies on two essential features:

1. First, as we shall see in what follows, the Galileon interactions or the interactions that arisein the decoupling limit of massive gravity and which are essential for the Vainshtein mecha-nism do not get renormalized within the decoupling limit (they enjoy a non-renormalizationtheorem which we review in what follows).

2. The non-renormalization theorem together with the shift and Galileon symmetry impliesthat only higher operators of the form

(𝜕ℓ𝜋)𝑚

, with ℓ,𝑚 ≥ 2 are generated by quantum



Massive Gravity 103

corrections. These operators differ from the Galileon operators in that they always generateterms that more than two derivatives on the field at the level of the equation of motion (orthey always have two or more derivatives per field at the level of the action).

This means that there exists a regime of interest for the theory, for which the operators gener-ated by quantum corrections are irrelevant (non-important compared to the Galileon interactions).Within the strong coupling region, the field itself can take large values, 𝜋 ∼ Λ, 𝜕𝜋 ∼ Λ2, 𝜕2𝜋 ∼ Λ3,and one can still rely on the Galileon interactions and take no other operator into account so longas any further derivative of the field is suppressed, 𝜕𝑛𝜋 ≪ Λ𝑛+1 for any 𝑛 ≥ 3.

This is similar to the situation in DBI scalar field models, where the field operator itself andits velocity is considered to be large 𝜋 ∼ Λ and 𝜕𝜋 ∼ Λ2, but the field acceleration and any higherderivatives are suppressed 𝜕𝑛𝜋 ≪ Λ𝑛+1 for 𝑛 ≥ 2 (see [157]). In other words, the Effective Fieldexpansion should be reorganized so that operators which do not give equations of motion withmore than two derivatives (i.e., Galileon interactions) are considered to be large and ought to betreated as the relevant operators, while all other interactions (which lead to terms in the equationsof motion with more than two derivatives) are treated as irrelevant corrections in the effective fieldtheory language.

Finally, as mentioned previously, the Vainshtein mechanism itself changes the canonical scaleand thus the scale at which the fluctuations become strongly coupled. On top of a backgroundconfiguration, interactions do not arise at the scale Λ but rather at the rescaled strong couplingscale Λ* =

√𝑍Λ, where 𝑍 is expressed in (10.7). In the strong coupling region, 𝑍 ≫ 1 and so

Λ* ≫ Λ. The higher interactions for fluctuations on top of the background configuration are hencemuch smaller than expected and their quantum corrections are therefore suppressed.

When taking the cubic Galileon and considering the strong coupling effect from a static andspherically symmetric source then

Λ* ∼√𝑍Λ ∼

√𝜋′0(𝑟)

𝑟Λ3Λ , (10.24)

where the profile for the cubic Galileon in the strong coupling region is given in (10.15). If thesource is considered to be the Earth, then at the surface of the Earth this gives

Λ* ∼(𝑀

𝑀Pl

1

(𝑟Λ)3

)Λ ∼ 107Λ ∼ cm−1 , (10.25)

taking Λ ∼ (1000 km)−1, which would be the scale Λ3 in massive gravity for a graviton mass ofthe order of the Hubble parameter today. In the quartic Galileon this enhancement in the strongcoupling scale does not work as well in the purely static and spherically symmetric case [88] howeverconsidering a more realistic scenario and taking the smallest breaking of the spherical symmetryinto account (for instance the Earth dipole) leads to a comparable result of a few cm [57]. Noticethat this is the redressed strong coupling scale when taking into consideration only the effect ofthe Earth. When getting to these smaller distance scales, all the other matter sources surroundingwhichever experiment or scattering process needs to be accounted for and this pushes the redressedstrong coupling scale even higher [57].

10.3 Non-renormalization

The non-renormalization theorem mentioned above states that within a Galileon theory the Galileonoperators themselves do not get renormalized. This was originally understood within the contextof the cubic Galileon in the procedure established in [411] and is easily generalizable to all theGalileons [412]. In what follows, we review the essence of non-renormalization theorem within thecontext of massive gravity as derived in [140].



104 Claudia de Rham

Let us start with the decoupling limit of massive gravity (8.38) in the absence of vector modes(the Vainshtein mechanism presented previously does not rely on these modes and it thus consistentfor the purpose of this discussion to ignore them). This decoupling limit is a very special scalar-tensor theory on flat spacetime

ℒΛ3= −1


1

4ℎ𝜇𝜈

3∑𝑛=1

𝑐𝑛

Λ3(𝑛−1)3

𝑋(𝑛)𝜇𝜈 , (10.26)

where the coefficients 𝑐𝑛 are given in (8.47) and the tensors 𝑋(𝑛) are given in (8.29 – 8.31) or(8.33 – 8.36). The theory described by (10.26) (including the two interactions ℎ𝑋(2,3)) enjoys twokinds of symmetries: a gauge symmetry for ℎ𝜇𝜈 (linearized diffeomorphism) ℎ𝜇𝜈 → ℎ𝜇𝜈 + 𝜕(𝜇𝜉𝜈)and a global shift and Galilean symmetry for 𝜋, 𝜋 → 𝜋 + 𝑐 + 𝑣𝜇𝑥

𝜇. Notice that unlike in a pureGalileon theory, here the global symmetry for 𝜋 is an exact symmetry of the Lagrangian (not asymmetry up to boundary terms). This means that the quantum corrections generated by thistheory ought to preserve the same kinds of symmetries.

The non-renormalization theorem follows simply from the antisymmetric structure of the in-teractions (8.30) and (8.31). Let us consider the contributions of the vertices

𝑉2 = ℎ𝜇𝜈𝑋(2)𝜇𝜈 = ℎ𝜇𝜈𝜀𝜇𝛼𝛽𝛾𝜀𝜈𝛼

′𝛽′

𝛾𝜕𝛼𝜕𝛼′𝜋𝜕𝛽𝜕𝛽′𝜋 (10.27)

𝑉3 = ℎ𝜇𝜈𝑋(3)𝜇𝜈 = ℎ𝜇𝜈𝜀𝜇𝛼𝛽𝛾𝜀𝜈𝛼

′𝛽′𝛾′𝜕𝛼𝜕𝛼′𝜋𝜕𝛽𝜕𝛽′𝜋𝜕𝛾𝜕𝛾′𝜋 (10.28)

to an arbitrary diagram. If all the external legs of this diagram are 𝜋 fields then it follows imme-diately that the contribution of the process goes as (𝜕2𝜋)𝑛 or with more derivatives and is thusnot an operator which was originally present in (10.26). So let us consider the case where a vertex(say 𝑉3) contributes to the diagram with a spin-2 external leg of momentum 𝑝𝜇. The contributionfrom that vertex to the whole diagram is given by

𝑖ℳ𝑉3∝ 𝑖∫

d4𝑘

(2𝜋)4d4𝑞

(2𝜋)4𝒢𝑘𝒢𝑞𝒢𝑝−𝑘−𝑞 (10.29)

×[𝜖*𝜇𝜈𝜀 𝛼𝛽𝛾

𝜇 𝜀 𝛼′𝛽′𝛾′

𝜈 𝑘𝛼𝑘𝛼′𝑞𝛽𝑞𝛽′(𝑝− 𝑘 − 𝑞)𝛾(𝑝− 𝑘 − 𝑞)𝛾′

]∝ 𝑖𝜖*𝜇𝜈𝜀 𝛼𝛽𝛾

𝜇 𝜀 𝛼′𝛽′𝛾′

𝜈 𝑝𝛾𝑝𝛾′

∫d4𝑘

(2𝜋)4d4𝑞

(2𝜋)4𝒢𝑘𝒢𝑞𝒢𝑝−𝑘−𝑞𝑘𝛼𝑘𝛼′𝑞𝛽𝑞𝛽′ ,

where 𝜖*𝜇𝜈 is the polarization of the spin-2 external leg and 𝒢𝑘 is the Feynman propagator for the𝜋-particle, 𝒢𝑘 = 𝑖(𝑘2 − 𝑖𝜀)−1. This contribution is quadratic in the momentum of the externalspin-2 field 𝑝𝛾𝑝𝛾′ , which means that in position space it has to involve at least two derivatives inℎ𝜇𝜈 (there could be more derivatives arising from the integral over the propagator 𝒢𝑝−𝑘−𝑞 insidethe loops). The same result holds when inserting a 𝑉2 vertex as explained in [140]. As a result anydiagram in this theory can only generate terms of the form (𝜕2ℎ)ℓ(𝜕2𝜋)𝑚, or terms with even morederivatives. As a result the operators presented in (10.26) or in the decoupling limit of massivegravity are not renormalized. This means that within the decoupling limit the scale Λ does notget renormalized, and it can be set to an arbitrarily small value (compared to the Planck scale)without running issues. The same holds for the other parameter 𝑐2 or 𝑐3.

When working beyond the decoupling limit, we expect operators of the form ℎ2(𝜕2𝜋)𝑛 to spoilthis non-renormalization theorem. However, these operators are 𝑀Pl suppressed, and so theylead to quantum corrections which are themselves 𝑀Pl suppressed. This means that the quantumcorrections to the graviton mass is suppressed as well [140]

𝛿𝑚2 . 𝑚2

(𝑚

𝑀Pl

)2/3

. (10.30)

This result is crucial for the theory. It implies that a small graviton mass is technically natural.



Massive Gravity 105

10.4 Quantum corrections beyond the decoupling limit

As already emphasized, the consistency of massive gravity relies crucially on a very specific set ofallowed interactions summarized in Section 6. Unlike for GR, these interactions are not protectedby any (known) symmetry and we thus expect quantum corrections to destabilize this structure.Depending on the scale at which these quantum corrections kick in, this could lead to a ghost atan unacceptably low scale.

Furthermore, as discussed previously, the mass of the graviton itself is subject to quantumcorrections, and for the theory to be viable the graviton mass ought to be tuned to extremelysmall values. This tuning would be technically unnatural if the graviton mass received largequantum corrections.

We first summarize the results found so far in the literature before providing further details

1. Destabilization of the potential:At one-loop, matter fields do not destabilize the structure of the potential. Graviton loops onthe hand do lead to new operators which do not belong to the ghost-free family of interactionspresented in (6.9 – 6.13), however they are irrelevant below the Planck scale.

2. Technically natural graviton mass:As already seen in (10.30), the quantum corrections for the graviton mass are suppressed bythe graviton mass itself, 𝛿𝑚2 . 𝑚2(𝑚/𝑀Pl)

2/3 this result is confirmed at one-loop beyondthe decoupling limit and as result a small graviton mass is technically natural.

10.4.1 Matter loops

The essence of these arguments go as follows: Consider a ‘covariant’ coupling to matter,ℒmatter(𝑔𝜇𝜈 , 𝜓𝑖), for any species 𝜓𝑖 be it a scalar, a vector, or a fermion (in which case the couplinghas to be performed in the vielbein formulation of gravity, see (5.6)).

At one loop, virtual matter fields do not mix with the virtual graviton. As a result as far asmatter loops are concerned, they are ‘unaware’ of the graviton mass, and only lead to quantumcorrections which are already present in GR and respect diffeomorphism invariance. So the onlypotential term (i.e., operator with no derivatives on the metric fluctuation) it can lead to is thecosmological constant.

This result was confirmed at the level of the one-loop effective action in [146], where it wasshown that a field of mass 𝑀 leads to a running of the cosmological constant 𝛿ΛCC ∼ 𝑀4. Thisresult is of course well-known and is at the origin of the old cosmological constant problem [484].The key element in the context of massive gravity is that this cosmological constant does notlead to any ghost and no new operators are generated from matter loops, at the one-loop level(and this independently of the regularization scheme used, be it dimensional regularization, cutoffregularization, or other.) At higher loops we expect virtual matter fields and graviton to mix andeffect on the structure of the potential still remains to be explored.

10.4.2 Graviton loops

When considering virtual gravitons running in the loops, the theory does receive quantum correc-tions which do not respect the ghost-free structure of the potential. These are of course suppressedby the Planck scale and the graviton mass and so in dimensional regularization, we generate new



106 Claudia de Rham

operators of the form25

ℒ(potential)QC ∼ 𝑚4

𝑀𝑛Pl

ℎ𝑛 , (10.31)

with 𝑛 ≥ 2, and where 𝑚 is the graviton mass, and the contractions of ℎ do not obey the structurepresented in (6.9) – (6.13). In a normal effective field theory this is not an issue as such operatorsare clearly irrelevant below the Planck scale. However, for massive gravity, the situation is moresubtle.

As see in Section 10.1 (see also Section 10.2), massive gravity is phenomenologically viable onlyif it has an active Vainshtein mechanism which screens the effect of the helicity-0 mode in thevicinity of dense environments. This Vainshtein mechanisms relies on having a large backgroundfor the helicity-0 mode, 𝜋 = 𝜋0 + 𝛿𝜋 with 𝜕2𝜋0 ≫ Λ3

3 = 𝑚2𝑀Pl, which in unitary gauge impliesℎ = ℎ0 + 𝛿ℎ, with ℎ0 ≫𝑀Pl.

To mimic this effect, we consider a given background for ℎ = ℎ0 ≫ 𝑀Pl. Perturbing thenew operators (10.31) about this background leads to a contribution at quadratic order for theperturbations 𝛿ℎ which does not satisfy the Fierz–Pauli structure,

ℒ(2)QC ∼

𝑚4 ℎ𝑛−20

𝑀𝑛Pl

𝛿ℎ2 . (10.32)

In terms of the helicity-0 mode 𝜋, considering 𝛿ℎ ∼ 𝜕2𝜋/𝑚2 this leads to higher derivative inter-actions

ℒ(2)QC ∼

ℎ𝑛−20

𝑀𝑛Pl

(𝜕2𝜋

)2, (10.33)

which revive the BD ghost at the scale 𝑚2ghost ∼ ℎ20(𝑀Pl/ℎ0)

𝑛. The mass of the ghost can be madearbitrarily small, (smaller than Λ3) by taking 𝑛≫ 1 and ℎ0 &𝑀Pl as is needed for the Vainshteinmechanism. In itself this would be a disaster for the theory as it means precisely in the regimewhere we need the Vainshtein mechanism to work, a ghost appears at an arbitrarily small scaleand we can no longer trust the theory.

The resolution to this issue lies within the Vainshtein mechanism itself and its implementationnot only at the classical level as was done to estimate the mass of the ghost in (10.33) but alsowithin the calculation of the quantum corrections themselves. To take the Vainshtein mechanismconsistently into account one needs to consider the effective action redressed by the interactionsthemselves (as was performed at the classical level for instance in (10.9)).

This redressing was taken into at the level of the one-loop effective action in Ref. [146] and it wasshown that when resumed, the large background configuration has the effect of further suppressingthe quantum corrections so that the mass of the ghost never reaches below the Planck scale evenwhen ℎ0 ≪𝑀Pl. To be more precise (10.33) is only one term in an infinite order expansion in ℎ0.Resuming these terms leads rather to contribution of the form (symbolically)

ℒ(2)QC ∼

1

1 + ℎ0

𝑀Pl

1

𝑀2Pl

(𝜕2𝜋

)2, (10.34)

so that the effective scale at which this operator is relevant is well above the Planck scale whenℎ0 & 𝑀Pl and is at the Planck scale when working in the weak-field regime ℎ0 . 𝑀Pl. Notice

25 This result has been checked explicitly in Ref. [146] using dimensional regularization or following the logdivergences. Taking power law divergences seriously would also allow for a scalings of the form (Λ4

Cutoff/𝑀𝑛Pl)ℎ

𝑛,which are no longer suppressed by the mass scale 𝑚 (although the mass scale 𝑚 would never enter with negativepowers at one loop.) However, it is well known that power law divergences cannot be trusted as they depend on themeasure of the path integral and can lead to erroneous results in cases where the higher energy theory is known. SeeRef. [84] and references therein for known examples and an instructive discussion on the use and abuses of powerlaw divergences.



Massive Gravity 107

that ℎ0 ∼ −𝑀Pl corresponds to a physical singularity in massive gravity (see [56]), and the theorywould break down at that point anyways, irrespectively of the ghost.

As a result, at the one-loop level the quantum corrections destabilize the structure of thepotential but in a way which is irrelevant below the Planck scale.

10.5 Strong coupling scale vs cutoff

Whether it is to compute the Vainshtein mechanism or quantum corrections to massive gravity, itis crucial to realize that the scale Λ = (𝑚2𝑀Pl)

1/3 (denoted as Λ in what follows) is not necessarilythe cutoff of the theory.

The cutoff of a theory corresponds to the scale at which the given theory breaks down andnew physics is required to describe nature. For GR the cutoff is the Planck scale. For massivegravity the cutoff could potentially be below the Planck scale, but is likely well above the scale Λ,and the redressed scale Λ* computed in (10.24). Instead Λ (or Λ* on some backgrounds) is thestrong-coupling scale of the theory.

When hitting the scale Λ or Λ* perturbativity breaks down (in the standard field representationof the theory), which means that in that representation loops ought to be taken into account toderive the correct physical results at these scales. However, it does not necessarily mean thatnew physics should be taken into account. The fact that tree-level calculations do not accountfor the full results does in no way imply that theory itself breaks down at these scales, only thatperturbation theory breaks down.

Massive gravity is of course not the only theory whose strong coupling scale departs from itscutoff. See, for instance, Ref. [31] for other examples in chiral theory, or in gravity coupled to manyspecies. To get more intuition on these types of theories and on the distinction between strongcoupling scale and cutoff, consider a large number 𝑁 ≫ 1 of scalar fields coupled to gravity. Inthat case the effective strong coupling scale seen by these scalars is 𝑀eff =𝑀Pl/

√𝑁 ≪𝑀Pl, while

the cutoff of the theory is still 𝑀Pl (the scale at which new physics enters in GR is independent ofthe number of species living in GR).

The philosophy behind [31] is precisely analogous to the distinction between the strong couplingscale and the cutoff (onset of new physics) that arises in massive gravity, and summarizing theresults of [31] would not make justice of their work, instead we quote the abstract and encouragethe reader to refer to that article for further details:

“In effective field theories it is common to identify the onset of new physics with theviolation of tree-level unitarity. However, we show that this is parametrically incorrectin the case of chiral perturbation theory, and is probably theoretically incorrect ingeneral. In the chiral theory, we explore perturbative unitarity violation as a functionof the number of colors and the number of flavors, holding the scale of the “new physics”(i.e., QCD) fixed. This demonstrates that the onset of new physics is parametricallyuncorrelated with tree-unitarity violation. When the latter scale is lower than thatof new physics, the effective theory must heal its unitarity violation itself, which isexpected because the field theory satisfies the requirements of unitarity. (. . . ) A similarexample can be seen in the case of general relativity coupled to multiple matter fields,where iteration of the vacuum polarization diagram restores unitarity. We presentarguments that suggest the correct identification should be connected to the onset ofinelasticity rather than unitarity violation.” [31].



108 Claudia de Rham

10.6 Superluminalities and (a)causality

Besides the presence of a low strong coupling scale in massive gravity (which is a requirementfor the Vainshtein mechanism, and is thus not a feature that should necessarily try to avoid),another point of concern is the possibility to have superluminal propagation. This statementsrequires a qualification and to avoid any confusion, we shall first review the distinction betweenphase velocity, group velocity, signal velocity and front velocity and their different implications.We follow the same description as in [399] and [77] and refer to these books and references thereinfor further details.

1. Phase Velocity: For a wave of constant frequency, the phase velocity is the speed at whichthe peaks of the oscillations propagate. For a wave [77]

𝑓(𝑡, 𝑥) = 𝐴 sin(𝜔𝑡− 𝑘𝑥) = 𝐴 sin

(𝜔

(𝑡− 𝑥

𝑣phase

)), (10.35)

the phase velocity 𝑣phase is given by

𝑣phase =𝜔

𝑘. (10.36)

2. Group Velocity: If the amplitude of the signal varies, then the group velocity representsthe speed at which the modulation or envelop of the signal propagates. In a medium wherethe phase velocity is constant and does not depend on frequency, the phase and the groupvelocity are the same. More generally, in a medium with dispersion relation 𝜔(𝑘), the groupvelocity is

𝑣group =𝜕𝜔(𝑘)

𝜕𝑘. (10.37)

We are familiar with the notion that the phase velocity can be larger than speed of light 𝑐(in this review we use units where 𝑐 = 1.) Similarly, it has been known for now almost acentury that

“(...) the group velocity could exceed 𝑐 in a spectral region of ananomalous dispersion” [399].

While being a source of concern at first, it is now well-understood not to be in any conflictwith the theory of general (or special) relativity and not to be the source of any acausality.The resolution lies in the fact that the group velocity does not represent the speed at whichnew information is transmitted. That speed is instead refer as the front velocity as we shallsee below.

3. Signal Velocity “yields the arrival of the main signal, with intensities of the order of mag-nitude of the input signal” [77]. Nowadays it is common to define the signal velocity as thevelocity from the part of the pulse which has reached at least half the maximum intensity.However, as mentioned in [399], this notion of speed rather is arbitrary and some knownphysical systems can exhibit a signal velocity larger than 𝑐.

4. Front Velocity: Physically, the front velocity represents the speed of the front of a distur-bance, or in other words “Front velocity (...) correspond[s] to the speed at which the very first,extremely small (perhaps invisible) vibrations will occur.” [77]. The front velocity is thus thespeed at which the very first piece of information of the first “forerunner” propagates once afront or a “sudden discontinuous turn-on of a field” is turned on [399].



Massive Gravity 109

“The front is defined as a surface beyond which, at a given instant in time the medium iscompletely at rest” [77],

𝑓(𝑡, 𝑥) = 𝜃(𝑡) sin(𝜔𝑡− 𝑘𝑥) , (10.38)

where 𝜃(𝑡) is the Heaviside step function.

In practise the front velocity is the large 𝑘 (high frequency) limit of the phase velocity.

The distinction between these four types of velocities in presented in Figure 5. They are importantto keep in mind and especially to be distinguished when it comes to superluminal propagation.Superluminal phase, group and signal velocities have been observed and measured experi-mentally in different physical systems and yet cause no contradiction with special relativity nor dothey signal acausalities. See Ref. [318] for an enlightening discussion of the case of QED in curvedspacetime.

The front velocity, on the other hand, is the real ‘measure’ of the speed of propagation ofnew information, and the front velocity is always (and should always be) (sub)luminal. As shownin [445], “the ‘speed of light’ relevant for causality is 𝑣ph(∞), i.e., the high-frequency limit of thephase velocity. Determining this requires a knowledge of the UV completion of the quantum fieldtheory.” In other words, there is no sense in computing a classical version of the front velocitysince quantum corrections always dominate.

When it comes to the presence of superluminalities in massive gravity and theories of Galileonsthis distinction is crucial. We first summarize the current state of the situation in the context ofboth Galileons and massive gravity and then give further details and examples in what follows:

∙ In Galileons theories the presence of superluminal group velocity has been established forall the parameters which exhibit an active Vainshtein mechanism. These are present inspherically symmetric configurations near massive sources as well as in self-sourced planewaves and other configurations for which no special kind of matter is required.

∙ Since massive gravity reduces to a specific Galileon theory in some limit, we expect the sameresult to be true there well and to yield solutions with superluminal group velocity. However,to date no fully consistent solution has yet been found in massive gravity which exhibitssuperluminal group velocity (let alone superluminal front velocity which would be the realsignal of acausality). Only local configurations have been found with superluminal groupvelocity or finite frequency phase velocity but it has not been proven that these are stableglobal solutions. Actually, in all the cases where this has been checked explicitly so far, theselocal configurations have been shown not to be part of global stable solutions.

It is also worth noting that the potential existence of superluminal propagation is not restrictedto theories which break the gauge symmetry. For instance, massless spin-3/2 are also known topropagate superluminal modes on some non-trivial backgrounds [306].

10.6.1 Superluminalities in Galileons

Superluminalities in Galileon and other closely related theories have been pointed out in severalstudies for more a while [412, 1, 262, 220, 115, 129, 246]. Note also that Ref. [313] was the firstwork to point out the existence of superluminal propagation in the higher-dimensional picture ofDGP rather than in its purely four-dimensional decoupling limit. See also Refs. [112, 110, 311, 312,218, 219] for related discussions on super- versus subluminal propagation in conformal Galileonand other DBI-related models. The physical interpretation of these superluminal propagations wasstudied in other non-Galileon models in [199, 43] and see [206, 469] for their potential connectionwith classicalization [214, 213, 205, 11].



110 Claudia de Rham

Figure 5: Difference between phase, group, signal and front velocities. At 𝑡 = 𝛿𝑡, the phase and groupvelocities are represented on the left and given respectively by 𝑣phase = 𝛿𝑥𝑃 /𝛿𝑡 and 𝑣group = 𝛿𝑥𝐺/𝛿𝑡 (in thelimit 𝛿𝑡 → 0.) The signal and front velocity represented on the right are given by 𝑣signal = 𝛿𝑥𝑆/𝛿𝑡 (where𝛿𝑥𝑆 is the point where at least half the intensity of the original signal is reached.) The front velocity isgiven by 𝑣front = 𝛿𝑥𝐹 /𝛿𝑡.

In all the examples found so far, what has been pointed out is the existence of a superluminalgroup velocity, which is the regime inspected is the same as the phase velocity. As we will seebelow (see Section 10.7), in the one example where we can compute the phase velocity for mo-menta at which loops ought to be taken into account, we find (thanks to a dual description) thatthe corresponding front velocity is exactly luminal even though the low-energy group velocity issuperluminal. This is no indication that all Galileon theories are causal but it comes to show howa specific Galileon theory which exhibits superluminal group velocity in some regime is dual to acausal theory.

In most of the cases considered, superluminal propagation was identified in a spherically sym-metric setting in the vicinity of a localized mass as was presented in Section 10.1.2. To convincethe reader that these superluminalities are independent of the coupling to matter, we show herehow superluminal propagation can already occur in the vacuum in any Galileon theories withouteven the need of any external matter.

Consider an arbitrary quintic Galileon

ℒ = 𝜋

4∑𝑛=1

𝑐𝑛+1

Λ3(𝑛−1)ℒ𝑛(Π) , (10.39)

where the ℒ𝑛 are given in (6.10) – (6.13) and we choose the canonical normalization 𝑐2 = 1/12.One can check that any plane-wave configuration of the form

𝜋0(𝑥𝜇) = 𝐹 (𝑥1 − 𝑡) , (10.40)

is a solution of the vacuum equations of motion for any arbitrary function 𝐹 ,

4∑𝑛=1

(𝑛+ 1)𝑐𝑛Λ3(𝑛−1)

ℒ𝑛(Π0) = 0 , (10.41)



Massive Gravity 111

with Π0𝜇𝜈 = 𝜕𝜇𝜕𝜈𝜋0, since ℒ𝑛(𝜕𝜇𝜕𝜈𝜋0) = 0 for any 𝑛 ≥ 1 for a plane-wave of the form (10.40).Now, considering perturbations riding on top of the plane-wave, 𝜋(𝑥𝜇) = 𝜋0(𝑡, 𝑥

1) + 𝛿𝜋(𝑥𝜇),these perturbations see an effective background-dependent metric similarly as in Section 10.1.1and have the linearized equation of motion

𝑍𝜇𝜈(𝜋0)𝜕𝜇𝜕𝜈𝛿𝜋 = 0 , (10.42)

with 𝑍𝜇𝜈 given in (10.7)

𝑍𝜇𝜈(𝜋0) =3∑

𝑛=0

(𝑛+ 1)(𝑛+ 2)𝑐𝑛+2

Λ3𝑛𝑋(𝑛)𝜇𝜈(Π0) (10.43)

=

[𝜂𝜇𝜈 − 12𝑐3

Λ3𝐹 ′′(𝑥1 − 𝑡)(𝛿𝜇0 + 𝛿𝜇1 )(𝛿

𝜈0 + 𝛿𝜈1 )

]. (10.44)

A perturbation traveling along the direction 𝑥1 has a velocity 𝑣 which satisfies

𝑍00𝑣2 + 2𝑍01𝑣 + 𝑍11 = 0 . (10.45)

So, depending on wether the perturbation travels with or against the flow of the plane wave, itwill have a velocity 𝑣 given by

𝑣 = −1 or 𝑣 =1− 12𝑐3

Λ3 𝐹′′(𝑥1 − 𝑡)

1 + 12𝑐3Λ3 𝐹 ′′(𝑥1 − 𝑡)

. (10.46)

So, a plane wave which admits26

12𝑐3𝐹′′ < −Λ3 , (10.47)

the perturbation propagates with a superluminal velocity. However, this velocity corresponds tothe group velocity and in order to infer whether or not there is any acausality we need to derive thefront velocity, which is the large momentum limit of the phase velocity. The derivation presentedhere presents a tree-level calculation and to compute the large momentum limit one would needto include loop corrections. This is especially important as 12𝑐3𝐹

′′ → −Λ3 as the theory becomes(infinitely) strongly coupled at that point [87]. So far, no computation has properly taken thesequantum effects into account, and the (a)causality of Galileons theories is yet to determined.

10.6.2 Superluminalities in massive gravity

The existence of superluminal propagation directly in massive gravity has been pointed out inmany references in the literature [87, 276, 192, 177] (see also [496] for another nice discussion).Unfortunately none of these studies have qualified the type of velocity which exhibits superluminalpropagation. On closer inspection it appears that there again for all the cases cited the superluminalpropagation has so far always been computed classically without taking into account quantumcorrections. These results are thus always valid for the low frequency group velocity but neverfor the front velocity which requires a fully fledged calculation beyond the tree-level classicalapproximation [445].

Furthermore, while it is very likely that massive gravity admits superluminal propagation, todate there is no known consistent solution of massive gravity which has been shown to admit super-luminal (even of group) velocity. We review the arguments in favor of superluminal propagation inwhat follows together with their limitations. Notice as well that while a Galileon theory typicallyadmits superluminal propagation on top of static and spherically symmetric Vainshtein solutionsas presented in Section 10.1.2, this is not the case for massive gravity see Section 10.1.3 and [58].

26 If 𝑐3 = 0, we can easily generalize the background solution to find other configurations that admit a superluminalpropagation.



112 Claudia de Rham

1. Argument: Some background solutions of massive gravity admit superluminalpropagation.Limitation of the argument: the solutions inspected were not physical.Ref. [276] was the first work to point out the presence of superluminal group velocity in thefull theory of massive gravity rather than in its Galileon decoupling limit. These superluminalmodes ride on top of a solution which is unfortunately unrealistic for different reasons. First,the solution itself is unstable. Second, the solution has no rest frame (if seen as a perfectfluid) or one would need to perform a superluminal boost to bring the solution to its restframe. Finally, to exist, such a solution should be sourced by a matter source with complexeigenvalues [142]. As a result the solution cannot be trusted in the first place, and so neithercan the superluminal propagation of fluctuations about it.

2. Argument: Some background solutions of the decoupling limit of massive gravityadmit superluminal propagation.Limitation of the argument: the solutions were only found in a finite region ofspace and time.In Ref. [87] superluminal propagation was found in the decoupling limit of massive gravity.These solutions do not require any special kind of matter, however the background has onlybe solved locally and it has not (yet) been shown whether or not they could extrapolate tosensible and stable asymptotic solutions.

3. Argument: There are some exact solutions of massive gravity for which thedeterminant of the kinetic matrix vanishes thus massive gravity is acausal.Limitation of the argument: misuse of the characteristics analysis – what hasreally been identified is the absence of BD ghost.Ref. [192] presented some solutions which appeared to admit some instantaneous modes inthe full theory of massive gravity. Unfortunately the results presented in [192] were due to amisuse of the characteristics analysis.

The confusion in the characteristics analysis arises from the very constraint that eliminatesthe BD ghost. The existence of such a constraint was discussed in length in many differentformulations in Section 7 and it is precisely what makes ghost-free (or dRGT) massive gravityspecial and theoretically viable. Due to the presence of this constraint, the characteristicsanalysis should be performed after solving for the constraints and not before [326].

In [192] it was pointed out that the determinant of the time kinetic matrix vanished in ghost-free massive gravity before solving for the constraint. This result was then interpreted asthe propagation of instantaneous modes and it was further argued that the theory was thenacausal. This result is simply an artefact of not properly taking into account the constraintand performing a characteristics analysis on a set of modes which are not all dynamical (sincetwo phase space variables are constrained by the primary and secondary constrains [295, 294]).In other word it is precisely what would–have–been the BD ghost which is responsible forcanceling the determinant of the time kinetic matrix. This does not mean that the BD ghostpropagates instantaneously but rather that the BD ghost is not present in that theory, whichis the very point of the theory.

One can show that the determinant of the time kinetic matrix in general does not vanishwhen computing it after solving for the constraints. In summary the results presented in [192]cannot be used to deduce the causality of the theory or absence thereof.

4. Argument: Massive gravity admits shock wave solutions which admit superlu-minal and instantaneous modes.



Massive Gravity 113

Limitation of the argument: These configurations lie beyond the regime of va-lidity of the classical theory.Shock wave local solutions on top of which the fluctuations are superluminal were foundin [177]. Furthermore, a characteristic analysis reveals the possibility for spacelike hyper-surfaces to be characteristic. While interesting, such configurations lie beyond the regime ofvalidity of the classical theory and quantum corrections ought to be included.

Having said that, it is likely that the characteristic analysis performed in [177] and then in[178] would give the same results had it been performed on regular solutions.27 This pointis discussed below.

5. Argument: The characteristic analysis shows that some field configurations ofmassive gravity admit superluminal propagation and the possibility for spacelikehypersurfaces to be characteristic.Limitation of the argument: Same as point 2. Putting this limitation aside thisresult is certainly correct classically and in complete agreement with previousresults presented in the literature (see point 2 where local solutions were given).Even though the characteristic analysis presented in [177] used shock wave local configura-tions, it is also valid for smooth wave solutions which would be within the regime of validityof the theory. In [178] the characteristic analysis for a shock wave was presented again andit was argued that CTCs were likely to exist.

To better see the essence behind the general characteristic analysis argument, let us look atthe (simpler yet representative) case of a Proca field with an additional quartic interactionas explored in [420, 467],

ℒ = −1

4𝐹 2𝜇𝜈 −

1

2𝑚2𝐴𝜇𝐴𝜇 −

1

4𝜆(𝐴𝜇𝐴𝜇)

2 . (10.48)

The idea behind the characteristic analysis is to “replace the highest derivative terms 𝜕𝑁𝐴by 𝑘𝑁𝐴” [420] so that one of the equations of motion is[

(𝑚2 + 𝜆𝐴𝜈𝐴𝜈)𝑘𝛼𝑘𝛼 + 2𝜆(𝐴𝜈𝑘𝜈)

2]𝑘𝜇𝐴𝜇 = 0 . (10.49)

When 𝜆 = 0, one can solve this equation maintaining 𝑘𝜇𝐴𝜇 = 0. Then there are certainlyfield configurations for which the normal to the characteristic surface is timelike and thus themode with 𝑘𝜇𝐴𝜇 = 0 can propagate superluminally in this Proca field theory. However, as weshall see below this very combination 𝒵 =

[(𝑚2 + 𝜆𝐴𝜈𝐴𝜈)𝑘

𝛼𝑘𝛼 + 2𝜆(𝐴𝜈𝑘𝜈)2]= 0 with 𝑘𝜇

timelike (say 𝑘𝜇 = (1, 0, 0, 0)) is the coefficient of the time-like kinetic term of the helicity-0mode. So one can never have

[(𝑚2 + 𝜆𝐴𝜈𝐴𝜈)𝑘

𝛼𝑘𝛼 + 2𝜆(𝐴𝜈𝑘𝜈)2]= 0 with 𝑘𝜇 = (1, 0, 0, 0) (or

any timelike direction) without automatically having an infinitely strongly helicity-0 modeand thus automatically going beyond the regime of validity of the theory (see Ref. [87] formore details.)

To see this more precisely, let us perform the characteristic analysis in the Stuckelberg lan-guage. An analysis performed in unitary gauge is of course perfectly acceptable, but toconnect with previous work in Galileons and in massive gravity the Stuckelberg formalism isuseful.

In the Stuckelberg language, 𝐴𝜇 → 𝐴𝜇 +𝑚−1𝜕𝜇𝜋, keeping track of the terms quadratic in

27 We thank the authors of [177, 178] for pointing this out.



114 Claudia de Rham

𝜋, we have

ℒ(2)𝜋 = −1

2𝑍𝜇𝜈𝜕𝜇𝜋𝜕𝜈𝜋 , (10.50)

with 𝑍𝜇𝜈 [𝐴𝜇] = 𝜂𝜇𝜈 +𝜆

𝑚2𝐴2𝜂𝜇𝜈 + 2

𝜆

𝑚2𝐴𝜇𝐴𝜈 . (10.51)

It is now clear that the combination found in the characteristic analysis 𝒵 is nothing otherthan

𝒵 ≡ 𝑍𝜇𝜈𝑘𝜇𝑘𝜈 , (10.52)

where 𝑍𝜇𝜈 is the kinetic matrix of the helicity-0 mode. Thus, a configuration with 𝒵 = 0 with𝑘𝜇 = (1, 0, 0, 0) implies that the 𝑍00 component of helicity-0 mode kinetic matrix vanishes.This means that the conjugate momentum associated to 𝜋 cannot be solved for in this time-slicing, or that the helicity-0 mode is infinitely strongly coupled.

This result should sound familiar as it echoes what has already been shown to happen inthe decoupling limit of massive gravity, or here of the Proca field theory (see [43, 468] forrelated discussions in that case). Considering the decoupling limit of (10.48) with 𝑚 → 0

and �� = 𝜆/𝑚4 → const, we obtain a decoupled massless gauge field and a scalar field,

ℒDL = −1

4𝐹 2𝜇𝜈 −

1

2(𝜕𝜋)2 − ��

4(𝜕𝜋)4 . (10.53)

For fluctuations about a given background configuration 𝜋 = 𝜋0(𝑥)+ 𝛿𝜋, the fluctuations seean effective metric 𝑍𝜇𝜈(𝜋0) given by

𝑍𝜇𝜈(𝜋0) =(1 + ��(𝜕𝜋0)

2)𝜂𝜇𝜈 + 2��𝜕𝜇𝜋0𝜕

𝜈𝜋0 . (10.54)

Of course unsurprisingly, we find 𝑍𝜇𝜈(𝜋0) ≡ 𝑚−2𝑍𝜇𝜈 [𝑚−1𝜕𝜇𝜋0]. The fact that we can findsuperluminal or instantaneous propagation in the characteristic analysis is equivalent to thestatement that in the decoupling limit there exists classical field configurations for 𝜋0 forwhich the fluctuations propagate superluminally (or even instantaneously). Thus, the resultsof the characteristic analysis are in agreement with previous results in the decoupling limitas was pointed out for instance in [1, 412, 87].

Once again, if one starts with a field configuration where the kinetic matrix is well defined,one cannot reach a region where one of the eigenvalues of 𝑍𝜇𝜈 crosses zero without goingbeyond the regime of validity of the theory as described in [87]. See also Refs. [318, 445] forthe use of the characteristic analysis and its relation to (micro-)causality.

The presence of instantaneous modes in some (self-accelerating) solutions of massive gravitywas actually pointed out from the very beginning. See Refs. [139] and [364] for an analysis ofself-accelerating solutions in the decoupling limit, and [125] for self-accelerating solutions in thefull theory (see also [264] for a complementary analysis of self-accelerating solutions.) All theseanalysis had already found instantaneous modes on some self-accelerating branches of massivegravity. However, as pointed out in all these analysis, the real question is to establish whether ornot these solutions lie within the regime of validity of the EFT, and whether one could reach suchsolutions with a finite amount of energy and while remaining within the regime of validity of theEFT.

This aspect connects with Hawking’s chronology protection argument which is already in effectin GR [302, 303], (see also [472] and [473] for a comprehensive review). This argument can beextended to Galileon theories and to massive gravity as was shown in Ref. [87].



Massive Gravity 115

It was pointed out in [87] and in many other preceding works that there exists local back-grounds in Galileon theories and in massive gravity which admit superluminal and instantaneouspropagation. (As already mentioned, in point 2. above in massive gravity it is however unclearwhether these localized backgrounds admit stable and consistent global realizations). The worrywith superluminal propagation is that it could imply the presence of CTCs (closed timelike curves).However, when ‘cranking up’ the background sufficiently so as to reach a solution which wouldadmit CTCs, the Galileon or the helicity-0 mode of the graviton becomes inevitably infinitelystrongly coupled. This means that the effective field theory used breaks down and the backgroundbecomes unstable with arbitrarily fast decay time before any CTC can ever be formed.

Summary: Several analyses have confirmed the existence of local configurations admitingsuperluminalities in massive gravity. At this point, we leave it to the reader’s discretion to decidewhether the existence of local classical configurations which admit superluminalities and sometimeseven instantaneous propagation means that the theory should be discarded. We bear in mind thefollowing considerations:

∙ No stable global solutions have been found with the same properties.

∙ No CTCs can been constructed within the regime of validity of the theory. As shown inRef. [87] CTCs constructed with these configurations always lie beyond the regime of validityof the theory. Indeed in order to create a CTC, a mode needs to become instantaneous. Assoon as a mode becomes instantaneous, the regime of validity of the classical theory is nulland classical considerations are thus obsolete.

∙ Finally, and most importantly, all the results presented so far for Galileons and massive grav-ity (including the ones summarized here), rely on classical configurations. As was explainedat the beginning of this section causality is determined by the front velocity for which classi-cal considerations break down. Therefore, no classical calculations can ever prove or disprovethe (a)causality of a theory.

10.6.3 Superluminalities vs Boulware–Deser ghost vs Vainshtein

We finish by addressing what would be an interesting connection between the presence of super-luminalities and the very constraint of massive gravity which removes the BD ghost which waspointed out in [192, 177, 178]. Actually, one can show that the presence of local configurationswhich admits superluminalities is generic to any theories of massive gravity, including DGP, cascad-ing gravity, non-Fierz–Pauli massive gravity and even other braneworld models and is not specificto the presence of a constraint which removes the BD ghost. For instance consider a theory ofmassive gravity for which the cubic interactions about flat spacetime different than that of theghost-free model of massive gravity. Then as shown in Section 2.5 (for instance, Eqs. (2.86) or2.89, see also [111, 173]) the decoupling limit analysis leads to terms of the form

ℒFP, 𝜋 = −1

2(𝜕𝜋)2 +

4

𝑀Pl𝑚4

([Π][Π2]− [Π3]

). (10.55)

As we have shown earlier, results from this decoupling limit are in full agreement with a charac-teristic analysis.

The plane wave solutions provided in (10.40) is still a vacuum solution in this case. Followingthe same analysis as that provided in Section 10.6.1, one can easily find modes propagating withsuperluminal group and phase velocity for appropriate choices of functions 𝐹 (𝑥1−𝑡) (while keepingwithin the regime of validity of the theory.)



116 Claudia de Rham

Alternatively, let us look a background configuration �� with Π = 𝜕2��. Without loss of generalityat any point 𝑥 one can diagonalize the matrix Π. Focusing on a mode traveling along the 𝑥1

direction with momentum 𝑘𝜇 = (𝑘0, 𝑘1, 0, 0), we find the dispersion relation(𝑘20 − 𝑘21

)+

16

𝑀Pl𝑚4(𝑘0 − 𝑘1)2

[𝑘21(Π0

0 + Π22 + Π3

3

)+ 𝑘20

(Π1

1 + Π22 + Π3

3

)+ 2𝑘0𝑘1Π

𝜇𝜇

](10.56)

= 0 .

The presence of higher power in 𝑘 is nothing else but the signal of the BD ghost about genericbackgrounds where Π = 0. Performing a characteristic analysis at this point would focus on thehigher powers in 𝑘 which are intrinsic to the ghost. One can follow instead the non-ghost modewhich is already present even when Π = 0. To follow this mode, it is therefore sufficient to performa perturbative analysis in 𝑘. Equation (10.56) can always be solved for 𝑘0 = 𝑘1 as well as for

𝑘0 = −𝑘1 +32

𝑀Pl𝑚4𝑘31(Π0

0 − Π11

)+𝒪

(𝑘51Π

2

𝑚8𝑀Pl

). (10.57)

We can, therefore, always find a configuration for which 𝑘20 > 𝑘21 at least perturbatively whichis sufficient to imply the existence of superluminalities. Even if this calculation was performedperturbatively, it still implies the presence classical superluminalities like in the previous analysisof Galileon theories or ghost-free massive gravity.

As a result the presence of local solutions in massive gravity which admit superluminalities isnot connected to the constraint that removes the BD ghost. Rather it is likely that the presenceof superluminalities could be tied to the Vainshtein mechanism (with flat asymptotic boundaryconditions), which as we have seen is crucial for these types of theories (see Refs. [1, 313] and[129] for a possible connection.) More recently, the presence of superluminalities has also beenconnected to the idea of classicalization which is tied to the Vainshtein mechanism [206, 469].It is possible that the only way these superluminalities could make sense is through this idea ofclassicalization. Needless to say this is very much speculative at the moment. Perhaps the Galileondualities presented below could help understanding these open questions.

10.7 Galileon duality

The low strong coupling scale and the presence of superluminalities raises the question of howto understand the theory beyond the redressed strong coupling scale, and whether or not thesuperluminalities are present in the front velocity.

A non-trivial map between the conformal Galileon and the DBI conformal Galileon was recentlypresented in [113] (see also [55]). The conformal Galileon side admits superluminal propagationwhile the DBI side of the map is luminal. Since both sides are related by a ‘simple’ field redefinitionwhich does not change the physics, and cannot change the causality of the theory, this suggeststhat the superluminalities encountered in that example must be in the group velocity rather thanthe front velocity.

Recently, another Galileon duality was proposed in [115] and [136] by use of simple Legendretransform. First encountered within the decoupling limit of bi-gravity [224], the duality can beseen as being related to the freedom in how to introduce the Stuckelberg fields. However, theduality survives independently from bi-gravity and could be significant in the context of massivegravity.

To illustrate this duality, we start with a full Galileon in 𝑑 dimensions as in (10.6)

𝑆 =

∫d𝑑𝑥

(𝜋

𝑑∑𝑛=1

𝑐𝑛+1

Λ3(𝑛−1)ℒ𝑛[Π]

), (10.58)



Massive Gravity 117

and perform the field redefinition

𝜋(𝑥) → 𝜌(��) = −𝜋(𝑥)− 1

2Λ3

(𝜕𝜋(𝑥)

𝜕𝑥𝜇

)2

(10.59)

𝑥𝜇 → ��𝜇 = 𝑥𝜇 + 𝜂𝜇𝜈1

Λ3

𝜕𝜋(𝑥)

𝜕𝑥𝜈, (10.60)

This transformation is fully invertible without requiring any inverse of derivatives,

𝜌(��) → 𝜋(𝑥) = −𝜌(��)− 1

2Λ3

(𝜕𝜌(��)

𝜕��𝜇

)2

(10.61)

��𝜇 → 𝑥𝜇 = ��𝜇 + 𝜂𝜇𝜈1

Λ3

𝜕𝜌(��)

𝜕��𝜈, (10.62)

so the field transformation is not non-local (at least not in the traditional sense) and does not hidedegrees of freedom.

In terms of the dual field 𝜌(��), the Galileon theory (10.58) is nothing other than anotherGalileon with different coefficients,

𝑆 =

∫d𝑑��

(𝜌(��)

𝑑∑𝑛=1

𝑝𝑛+1

Λ3(𝑛−1)ℒ𝑛[Σ]

), (10.63)

with Σ𝜇𝜈 = 𝜕2𝜌(��)/𝜕��𝜇𝜕��𝜈 and the new coefficients are given by [136]

𝑝𝑛 =1

𝑛

𝑑+1∑𝑘=2

(−1)𝑘𝑐𝑘𝑘(𝑑− 𝑘 + 1)!

(𝑛− 𝑘)!(𝑑− 𝑛+ 1)!. (10.64)

This duality thus maps a Galileon to another Galileon theory with different coefficients. In partic-ular this means that the free theory 𝑐𝑛>2 = 0 maps to another non-trivial (𝑑+1)th order Galileontheory with 𝑝𝑛 = 0 for any 2 ≤ 𝑛 ≤ 𝑑 + 1. This dual Galileon theory admits superluminal prop-agation precisely in the same way as was pointed out on the spherically symmetric configurationsof Section 10.1.2 or on the plane wave solutions of Section 10.6.1. Yet, this non-trivial Galileon isdual to a free theory which is causal and luminal by definition.

What was computed in these examples for a non-trivial Galileon theory (and in all the examplesknown so far in the literature) is only the tree-level group velocity valid till the (redressed) strongcoupling scale of the theory. Once hitting the (redressed) strong coupling scale the loops need tobe included. In the dual free theory however there are no loops to account for, and thus the resultof luminal velocity in that free theory is valid at all scale and has to match the front velocity.This is strongly suggestive that the front velocity in that example of non-trivial Galileon theory isluminal and the theory is causal even though it exhibits a superluminal group velocity.

It is clear at this point that a deeper understanding of this class of theories is required. Weexpect this will be the subject of further studies. In the rest of this review, we focus on somephenomenological aspects of massive gravity before presenting other theories of massive gravity.



118 Claudia de Rham

Part III

Phenomenological Aspects of Ghost-freeMassive Gravity

11 Phenomenology

Below, we summarize some of the phenomenology of massive gravity and DGP. Many other in-teresting results have been derived in the literature, including the implication for the very EarlyUniverse. For instance false vacuum decay and the Hartle–Hawking no-boundary proposal wasstudied in the context of massive gravity in [498, 439, 499] where it was shown that the gravitonmass could increase the rate. The implications of massive gravity to the cyclic Universe were alsostudied in Ref. [91] with a regular bounce. We emphasize that in massive gravity the referencemetric has to be chosen once and for all and cannot be modified, no matter what the backgroundconfiguration is (no matter whether we are interested in cosmology, or in spherically symmetryconfigurations or other). This is a consistent procedure since massive gravity has been shownto be free of the BD ghost for any choice of reference metric independently of the backgroundconfiguration. Theories with different reference metrics represent different independent theories.

11.1 Gravitational waves

11.1.1 Speed of propagation

If the photon had a mass it would no longer propagate at ‘the speed of light’, but at a lower speed.For the photon its speed of propagation is known with such an accuracy in so many different mediathat it can be used to put the most stringent constraints on the photon mass to [68]𝑚𝛾 < 10−18 eV.In the rest of this review we will adopt the viewpoint that the photon is massless and that lightdoes indeed propagate at the ‘speed of light’.

The earliest bounds on the graviton mass were based on the same idea. As described in [487],(see also [394]), if the graviton had a mass, gravitational waves would propagate at a speed differentthan that of light, 𝑣2𝑔 = 1−𝑚2/𝐸2 (assuming a speed of light 𝑐 = 1). This different velocity betweenthe light and gravitational waves would manifest itself in observations of supernovae. Assuming theemission of a gravitational wave with frequency larger than the graviton mass, this could lead to abound on the graviton mass of 𝑚 < 10−23 eV considering a frequency of 100 Hz and a supernovaelocated 200 Mpc away [487] (assuming that the photon propagates at the speed of light).

Alternatively, another way to test the speed of gravitational waves and bound the gravitonmass without relying on any assumptions on the photon is through the observation of inspirallingcompact objects which allows to derive the frequency-dependence of GWs. The detection of GWsin Advanced LIGO could then bound the graviton mass potentially all the way down to 𝑚 <10−29 eV [487, 486, 71].

The graviton mass is also relevant for the production of primordial gravitational waves duringinflation. Following the analysis of [282] it was shown that the graviton mass opens up the pro-duction of gravitational waves during inflation with a sharp peak with a height and position whichdepend on the graviton mass. See also [403] for the study of exact plane wave solutions in massivegravity.

Nevertheless, these bounds on the graviton mass are relatively weak compared to the typicalvalue of 𝑚 ∼ 10−30 – 10−33 eV considered till now in this review. The reason for this is becausethese bounds do not take into account the effects arising from the additional polarization in thegravitational waves which would be present if the graviton had a mass in a Lorentz-invariant theory.



Massive Gravity 119

For the photon, if it had a mass, the additional polarization would decouple and would thereforebe irrelevant (this is related to the absence of vDVZ discontinuity at the classical level for a Procatheory.) In massive gravity, however, the helicity-0 mode of the graviton couples to matter. Aswe shall see below, the bounds on the graviton mass inferred from the absence of fifth forces aretypically much more stringent.

11.1.2 Additional polarizations

One of the predictions of GR is the existence of gravitational waves (GW) with two transverseindependent polarizations.

While GWs have not been directly detected via interferometer yet, they have been detectedthrough the spin-down of binary pulsar systems [322, 457, 485]. This detection via binary pulsarsdoes not count as a direct detection, but it matches expectations from GWs with such an accuracy,and for now so many different systems of different relative masses that it seems unlikely that thespin-down could be due to something different than the emission of GWs.

In a modified theory of gravity, one could expect a total of up to six polarizations for the GWsas seen in Figure 6.

Tensor mode

Tensor mode

Polarizations present in GR: Fully transverse to the line of propagation

Vector mode 1, 2

Scalar mode 1

Conformal mode

Scalar mode 2

Longitudinal mode

Additional Polarizations not present in GR

Figure 6: Polarizations of gravitational waves in general relativity and potential additional polarizationsin modified gravity.



120 Claudia de Rham

As emphasized in the first part of this review, and particularly in Section 2.5, the sixth exci-tation, namely the longitudinal one, represents a ghost degree of freedom. Thus, if that mode isobserved, it cannot be arising from a Lorentz-invariant massive graviton. Its presence could belinked for instance to new scalar degrees of freedom which are independent from the graviton itself.In massive gravity, only five polarizations are expected. Notice however that the helicity-1 modedoes not couple directly to matter or external sources, so it is unlikely that GWs with polarizationswhich mix the transverse and longitudinal directions would be produced in a natural process.

Furthermore, any physical process which is expected to produce GWs would include very densesources where the Vainshtein mechanism will thus be expected to be active and screen the effect ofthe helicity-0 mode. As a result the excitation of the breathing mode is expected to be suppressedin any theory of massive gravity which includes an active Vainshtein mechanism.

So, while one could in principle expect up to six polarizations for GWs in a modified theoryof gravity, in massive gravity only the two helicity-2 polarizations are expected to be produced ina potentially observable amount by interferometers like advanced-LIGO [289]. To summarize, inghost-free massive gravity or DGP we expect the following:

∙ The helicity-2 modes are produced in the same way as in GR and would be indistinguishableif they travel distances smaller than the graviton Compton wavelength

∙ The helicity-1 modes are not produced

∙ The breathing or conformal mode is produced but suppressed by the Vainshtein mechanismand so the magnitude of this mode is suppressed compared to the helicity-2 polarization bymany orders of magnitudes.

∙ The longitudinal mode does not exist in a ghost-free theory of massive gravity. If such amode is observed it must be arise from another field independent from the graviton.

We will also discuss the implications for indirect detection of GWs via binary pulsar spin-downin Section 11.4. We will see that already in these setups the radiation in the breathing mode issuppressed by 8 orders of magnitude compared to that in the helicity-2 mode. In more relativisticsystems such as black-hole mergers, this suppression will be even bigger as the Vainshtein mecha-nism is stronger in these cases, and so we do not expect to see the helicity-0 mode component ofa GW emitted by such systems.

To summarize, while additional polarizations are present in massive gravity, we do not expectto be able to observe them in current interferometers. However, these additional polarizations,and in particular the breathing mode can have larger effects on solar-system tests of gravity (seeSection 11.2) as well as for weak lensing (see Section 11.3), as we review in what follows. Theyalso have important implications for black holes as we discuss in Section 11.5 and in cosmology inSection 12.

11.2 Solar system

A lot of the phenomenology of massive gravity can be derived from its decoupling limit whereit resembles a Galileon theory. Since the Galileon was first encountered in DGP most of thephenomenology was first derived for that model. The extension to massive gravity is usuallyrelatively straightforward with a few subtleties which we mention at the end. We start by reviewingthe phenomenology assuming a cubic Galileon decoupling limit, which is directly applicable forDGP and then extend to the quartic Galileon and ghost-free massive gravity.

Within the context of DGP, a lot of its phenomenology within the solar system was derivedin [388, 386] using the full higher-dimensional picture as well as in [215]. In these work the effectfrom the helicity-0 mode in the advanced of the perihelion were computed explicitly. In particular



Massive Gravity 121

in [215] it was shown how an infrared modification of gravity could have an effect on small solarsystem scales and in particular on the Moon. In what follows we review their approach.

Consider a point source of mass 𝑀 localized at 𝑟 = 0. In GR (or rather Newtonian gravity asit is a sufficient approximation), the gravitational potential mediated by the point source is

Ψ(𝑟) = −ℎ00 = − 𝑀

4𝜋𝑀Pl

1

𝑟= − 1

𝑀Pl

𝑟𝑆𝑟, (11.1)

where 𝑟𝑆 is the Schwarzschild radius associated with the source. Now, in a theory of massivegravity, the helicity-0 mode of the graviton also contributes to the gravitational potential withan additional amount 𝛿Ψ. As seen in Section 10.1, when the Vainshtein mechanism is active thecontribution from the helicity-0 mode is very much suppressed. 𝛿Ψ ≪ Ψ but measurements inthe Solar system are reaching such a level of accuracy than even a small deviation 𝛿Ψ could inprinciple be observable [488].

In the decoupling limit of DGP, matter fields couple to the following perturbed metric

ℎDGP𝜇𝜈 = ℎEinstein

𝜇𝜈 + 𝜋0𝜂𝜇𝜈 , (11.2)

where 𝜋 is the helicity-0 mode of the graviton (up to some dimensionless numerical factors whichwe have set to unity). In massive gravity, matter couples to the following metric (see the discussionin Section 10.1.3 and (10.21)),

ℎmassive gravity𝜇𝜈 = ℎEinstein

𝜇𝜈 + 𝜋0𝜂𝜇𝜈 +𝛼

Λ33

𝜕𝜇𝜋0𝜕𝜈𝜋0 . (11.3)

The deviation 𝛿Ψ to the gravitational potential is thus given by

𝛿Ψ = −𝜋0 , (11.4)

(notice that in the static and spherically symmetric case 𝜕𝜇𝜋𝜕𝜈𝜋 leads to no correction to thegravitational potential).

Following [215] we define as 𝜖 the fractional change in the gravitational potential

𝜖(𝑟) =𝛿Ψ

Ψ=𝜋0(𝑟)

𝑀Pl

𝑟

𝑟𝑆. (11.5)

This change in the Newtonian force implies a change in the motion of a test particle (for instancethe Moon) within that gravitational field of the localized mass 𝑀 (of for instance the Earth) ascompared to GR. For elliptical orbits this leads to an additional angular precession of the periheliondue to the force mediated by the helicity-0 mode on top of that of GR. The additional advancedof the perihelion per orbit is given in terms of 𝜖 as

𝛿𝜑 = 𝜋𝑅0d

d𝑟

(𝑟2

d

d𝑟(𝑟−1𝜖)

) 𝑅0

, (11.6)

where 𝑅0 is the mean orbit radius, (notice the 𝜋 in that expression is the standard value 𝜋 = 3.14 . . .nothing to do with the helicity-0 mode).

DGP and cubic Galileon

In the decoupling limit of DGP (cubic Galileon) 𝜋 was given in (10.15) and 𝑟−1𝜖 ∼ (𝑟/𝑟3*)1/2,

where 𝑟* is the strong coupling radius derived in (10.14), 𝑟* = Λ−13 (𝑀/4𝜋𝑀Pl)

1/3 leading to ananomalous advance of the perihelion

𝛿𝜑 ∼ 3𝜋

4

(𝑟

𝑟*

)3/2

. (11.7)



122 Claudia de Rham

When the graviton mass goes to zero Λ3 → ∞ and the departure from GR goes to zero, thisis another example of how the Vainshtein mechanism arises. Interestingly, it was pointed outin [388, 386] that in DGP the sign of this anomalous angle depends on whether on the branchstudied (self-accelerating branch – or normal branch).

For the Earth-Moon system, taking Λ33(𝑚

2𝑀Pl) with 𝑚 ∼ 𝐻0 ∼ 10−33 eV, this leads to ananomalous precision of the order of [215]

𝛿𝜑 ∼ 10−12 rad/orbit , (11.8)

which is just on the edge of the level of accuracy currently reached by the lunar laser rangingexperiment [488] (for instance the accuracy quoted for the effective variation of the Gravitationalconstant is (4± 9)× 10−13/year ∼ (0.5± 1)× 10−11/orbit).

As pointed out in [215] and [388, 386], the effect could be bigger for the advance of the perihelionof Mars around the Sun, but at the moment the accuracy is slightly less.

Massive gravity and quartic Galileon:

As already mentioned in Section 10.1.2, the Vainshtein mechanism is typically much stronger28 inthe spherically symmetric configuration of the quartic Galileon and thus in massive gravity (seefor instance the suppression of the force given in (10.17)). Using the same values as before for aquartic Galileon we obtain

𝛿𝜑 ∼ 2𝜋

(𝑟

𝑟*

)2

∼ 10−16 /orbit . (11.9)

Furthermore, in massive gravity the parameter that enters this relation is not directly the gravitonbut rather the graviton mass weighted with the coefficient 𝛼 = −(1 + 3/2𝛼3) which depends onthe cubic potential term ℒ3, assuming that 𝛼4 = −𝛼3/4, (see Section 10.1.3 for more precision)

𝛿𝜑 ∼ 10−16

(1

𝛼

𝑚2

(10−33 eV)2

)2/3

/orbit , (11.10)

This is typically very far from observations unless we are very close to the minimal model.29

11.3 Lensing

As mentioned previously, one peculiarity of massive gravity not found in DGP nor in a typicalGalileon theory (unless we derive the Galileons from a higher-dimensional brane picture [157]) isthe new disformal coupling to matter of the form 𝜕𝜇𝜋𝜕𝜈𝜋𝑇

𝜇𝜈 , which means that the helicity-0mode also couples to conformal matter.

In the vacuum, for a static and spherically symmetric configuration the coupling 𝜕𝜇𝜋𝜕𝜈𝜋𝑇𝜇𝜈

plays no role. So to the level at which we are working when deriving the Vainshtein mechanismabout a point-like mass this additional coupling to matter does not affect the background con-figuration of the field (see [140] for a discussion outside the vacuum, taking into account for theinstance the effect of the Earth atmosphere). However, it does affect this disformal coupling doesaffect the effect metric seen by perturbed sources on top of this configuration. This could havesome implications for structure formation is to the best of our knowledge have not been fully ex-plored yet, and does affect the bending of light. This effect was pointed out in [490] and the effects

28 This is not to say that perturbations and/or perturbativity do not break down earlier in the quartic Galileon,see for instance Section 11.4 below as well as [88, 58], which is another sign that the Vainshtein mechanism worksbetter in that case.

29 The minimal model does not have a Vainshtein mechanism [435] in the static and spherically symmetricconfiguration so in the limit 𝛼3 → −1/3, or equivalently 𝛼 → 0, we indeed expect an order one correction.



Massive Gravity 123

to gravitational lensing were explored. We review the key results in what follows and refer to [490]for further discussions (see also [448]).

In GR, the relevant potential for lensing is Φ𝐿 = 12 (Φ−Ψ) ∼ 2 (𝑟𝑆/𝑟), where we use the same

notation as before, ℎ00 = Ψ and ℎ𝑖𝑗 = Φ𝛿𝑖𝑗 . A conformal coupling of the form 𝜋𝜂𝜇𝜈 does not affectthis lensing potential but the disformal coupling 𝛼/Λ3

3𝑀Pl𝜕𝜇𝜋𝜕𝜇𝜋𝑇𝜇𝜈 leads to a new contribution

𝛿Φ given by

𝛿Φ =𝛼

Λ33𝑀Pl

𝜋′0(𝑟)

2 . (11.11)

[Note we use a different notation that in [490], here 𝛼 = 1 + 3𝛼3.] This new contribution to thelensing potential leads to an anomalous fractional lensing of

ℛ =12𝛿Φ

Φ𝐿∼ 𝑟

4𝑟𝑆

(Λ3

𝛼1/3𝑀Pl

)(𝑀

4𝜋𝑀Pl

)2/3

. (11.12)

For the bending of light about the Sun, this leads to an effect of the order of

ℛ ∼ 10−11

(1

𝛼

𝑚2

(10−33eV)2

)1/3

, (11.13)

which is utterly negligible. Note that this is a tree-level calculation. When getting at thesedistances loops ought to be taken into account as well.

At the level of galaxies or clusters of galaxy, the effect might be more tangible. The reason forthat is that for the mass of a galaxy, the associated strong coupling radius is not much larger thanthe galaxy itself and thus at the edge of a galaxy these effects could be stronger. These effects wereinvestigated in [490] where it was shown a few percent effect on the tangential shear caused by thehelicity-0 mode of the graviton or of a disformal Galileon considering a Navarro–Frenk–White haloprofile, for some parameters of the theory. Interestingly, the effect peaks at some specific radiuswhich is the same for any halo when measured in units of the viral radius. Even though the effectis small, this peak could provide a smoking gun for such modifications of gravity.

Recently, another analysis was performed in Ref. [407], where the possibility to testing theoriesof modified gravity exhibiting the Vainshtein mechanism against observations of cluster lensing wasexplored. In such theories, like in massive gravity, the second derivative of the field can be largeat the transition between the screened and unscreened region, leading to observational signaturesin cluster lensing.

11.4 Pulsars

One of the main predictions of massive gravity is the presence of new polarizations for GWs.While these new polarization might not be detectable in GW interferometers as explained inSection 11.1.2, we could still expect them to lead to detectable effects in the binary pulsar systemswhose spin-down is in extremely good agreement with GR. In this section, we thus consider thepower emitted in the helicity-0 mode of the graviton in a binary-pulsar system. We use the effectiveaction approach derived by Goldberger and Rothstein in [254] and start with the decoupling limitof DGP before exploring that of ghost-free massive gravity and discussing the subtleties that arisein that case. We mainly focus on the monopole and quadrupole radiation although the wholeformalism can be derived for any multipoles. We follow the derivation of Refs. [158, 151], see alsoRefs. [100, 18] for related studies.

In order to account for the Vainshtein mechanism into account we perform a similar background-perturbation split as was performed in Section 10.1. The source is thus split as 𝑇 = 𝑇0+ 𝛿𝑇 where𝑇0 is a static and spherically source representing the total mass localized at the center of mass and𝛿𝑇 captures the motion of the companions with respect to the center of mass.



124 Claudia de Rham

This matter profile leads to a profile for the helicity-0 mode (here mimicked as a cubic Galileonwhich is the case for DGP) as in (10.3) as 𝜋 = 𝜋0(𝑟)+𝜑, where the background 𝜋0(𝑟) has the samestatic and spherical symmetry as 𝑇0 and so has the same profile as in Section 10.1.2.

The background configuration 𝜋0(𝑟) of the field was derived in (10.13) where𝑀 accounts in thiscase for the total mass of both companions and 𝑟 is the distance to the center of mass. Followingthe same procedure, the fluctuation 𝜑 then follows a modified Klein–Gordon equation

𝑍(𝜋0)𝜕2𝑥𝜑(𝑥) = 0 , (11.14)

where the Vainshtein mechanism is fully encoded in the background dependent prefactor 𝑍(𝜋0) ∼1 + 𝜕2𝜋0/Λ

3 and 𝑍(𝜋0) ≫ 1 in the vicinity of the binary pulsar system (well within the strongcoupling radius defined in (10.14).)

Expanding the field in spherical harmonics the mode functions satisfy

𝑍(𝜋0)𝜕2𝑥

[𝑢ℓ(𝑟)𝑌ℓ𝑚(Ω)𝑒−𝑖𝜔𝑡

]= 0 , (11.15)

where the modes are normalized so as to satisfy the standard normalization in the WKB region,for 𝑟 ≫ 𝜔−1.

The total power emitted via the field 𝜋 is given by the sum over these mode functions,

𝑃 (𝜋) =∞∑ℓ=0

𝑃(𝜋)ℓ =

∞∑ℓ=0

ℓ∑𝑚=−ℓ

∑𝑛≥0

(𝑛Ω𝑃 ) 1

𝑀Pl𝑇𝑃

∫ 𝑇𝑃

0

d4𝑥𝑢ℓ(𝑟)𝑌ℓ,𝑚𝑒−𝑖𝑛Ω𝑃 𝑡𝛿𝑇

2, (11.16)

where 𝑇𝑃 is the orbital period of the binary system and Ω𝑃 = 2𝜋/𝑇𝑝 is the corresponding angular

velocity. 𝑃(𝜋)0 is the power emitted in the monopole, 𝑃

(𝜋)1 in the dipole 𝑃

(𝜋)2 in the quadrupole of

the field 𝜋 uniquely, etc. . . in addition to the standard power emitted in the helicity-2 quadrupolechannel of GR.

Without the Vainshtein mechanism, the mode functions would be the same as for a standardfree-field in flat space-time, 𝑢ℓ ∼ 1

𝑟√𝜋𝜔

cos(𝜔𝑟) and the power emitted in the monopole would be

larger than that emitted in GR, which would be clearly ruled out by observations. The Vainshteinmechanism is thus crucial here as well for the viability of DGP or ghost-free massive gravity.

Monopole

Taking the prefactor 𝑍(𝜋0) into account, the zero mode for the monopole is given instead by

𝑢0(𝑟) ∼1

(𝜔𝑟3*)1/4

(1− (𝜔𝑟)2

4+ · · ·

), (11.17)

in the strong coupling regime 𝑟 ≪ 𝜔−1 ≪ 𝑟* which is the region where the radiation would beemitted. As a result, the power emitted in the monopole channel through the field 𝜋 is givenby [158]

𝑃(𝜋)0 = 𝜅

(Ω𝑃 𝑟)4

(Ω𝑃 𝑟*)3/2ℳ2

𝑀2Pl

Ω2𝑃 , (11.18)

where ℳ is the reduced mass and 𝑟 is the semi-major axis of the orbit and 𝜅 is a numericalprefactor of order 1 which depends on the eccentricity of the orbit.

This is to be compared with the Peters–Mathews formula for the power emitted in GR (in thehelicity-2 modes) in the quadrupole [428],

𝑃(Peters−Mathews)2 = �� (Ω𝑃 𝑟)

4 ℳ2

𝑀2Pl

Ω2𝑃 , (11.19)



Massive Gravity 125

where �� is again a different numerical prefactor which depends on the eccentricity of the orbit,and ℳ is a different combination of the companion masses, when both masses are the same (as isalmost the case for the Hulse–Taylor pulsar),ℳ = ℳ.

We see that the radiation in the monopole is suppressed by a factor of (Ω𝑃 𝑟*)−3/2 compared

with the GR result. For the Hulse–Taylor pulsar this is a suppression of 10 orders of magnitudeswhich is completely unobservable (at best the precision of the GR result is of 3 orders of magnitude).

Notice, however, that the suppression is far less than what was naively anticipated from thestatic approximation in Section 10.1.2.

The same analysis can be performed for the dipole emission with an even larger suppression ofabout 19 orders of magnitude compared the Peters–Mathews formula.

Quadrupole

The quadrupole emission in the field 𝜋 is slightly larger than the monopole. The reason is thatenergy conservation makes the non-relativistic limit of the monopole radiation irrelevant and oneneeds to take the first relativistic correction into account to emit in that channel. This is not sofor the quadrupole as it does not correspond to the charge associated with any Noether currenteven in the non-relativistic limit.

In the non-relativistic limit, the mode function for the quadrupole is simply𝑢2(𝑟) ∼ (𝜔𝑟)3/2/(𝜔𝑟3*)

1/4 yielding a quadrupole emission

𝑃(𝜋)2 = ��

(Ω𝑃 𝑟)3

(Ω𝑃 𝑟*)3/2ℳ2

𝑀2Pl

Ω2𝑃 , (11.20)

where �� is another numerical factor which depends on the eccentricity of the orbit and ℳ anotherreduced mass. The Vainshtein suppression in the quadrupole is (Ω𝑃 𝑟*)

−3/2(Ω𝑟)−1 ∼ 10−8 for theHulse–Taylor pulsar, and is thus well below the limit of being detectable.

Quartic Galileon

When extending the analysis to more general Galileons or to massive gravity which includes aquartic Galileon, we expect a priori by following the analysis of Section 10.1.2, to find a strongerVainshtein suppression. This result is indeed correct when considering the power radiated inonly one multipole. For instance in a quartic Galileon, the power emitted in the field 𝜋 via thequadrupole channel is suppressed by 12 orders of magnitude compared the GR emission.

However, this estimation does not account for the fact that there could be many multipolescontributing with the same strength in a quartic Galileon theory [151].

In a quartic Galileon theory, the effective metric in the strong coupling radius for a static andspherically symmetric background is

𝑍𝜇𝜈 d𝑥𝜇 d𝑥𝜈 ∼(𝜋′0

Λ3𝑟

)2 (− d𝑡2 + d𝑟2 + 𝑟2* dΩ2

), (11.21)

the fact that the angular direction is not suppressed by 𝑟2 but rather by a constant 𝑟2* implies thatthe multipoles are no longer suppressed by additional powers of velocity as is the case in GR or inthe cubic Galileon. This implies that many multipoles contribute with the same strength, yieldinga potentially large results. This is a sign that perturbation theory is not under control on top ofthis static and spherically symmetric background and one should really consider a more realisticbackground which will resume some of these contributions.

In situations where there is a large hierarchy between the mass of the two objects (which isthe case for instance within the solar system), perturbation theory can be seen to remain undercontrol and the power emitted in the quartic Galileon is completely negligible.



126 Claudia de Rham

11.5 Black holes

As in any gravitational theory, the existence and properties of black holes are crucially importantfor probing the non-perturbative aspects of gravity. The celebrated black-hole theorems of GR playa significant role in guiding understanding of non-perturbative aspects of quantum gravity. Fur-thermore, the phenomenology of black holes is becoming increasingly important as understandingof astrophysical black holes increases.

Massive gravity and its extensions certainly exhibit black-hole solutions and if the Vainshteinmechanism is successful then we would expect solutions which look arbitrary close to the Schwarz-schild and Kerr solutions of GR. However, as in the case of cosmological solutions, the situation ismore complicated due to the absence of a unique static spherically symmetric solution that arisesfrom the existence of additional degrees of freedom, and also the existence of other branches ofsolutions which may or may not be physical. There are a handful of known exact solutions inmassive gravity [413, 363, 365, 277, 105, 56, 477, 90, 478, 455, 30, 357], but the most interestingand physically relevant solutions probably correspond to the generic case where exact analyticsolutions cannot be obtained. A recent review of black-hole solutions in bi-gravity and massivegravity is given in [478].

An interesting effect was recently found in the context of bi-gravity in Ref. [41]. In thatcase, the Schwarzschild solutions were shown to be unstable (with a Gregory–Laflamme type ofinstability [268, 269]) at a scale dictated by the graviton mass, i.e., the instability rate is of theorder of the age of the Universe. See also Ref. [42] where the analysis was generalized to the non-bidiagonal. In this more general situation, spherically symmetric perturbations were also found butgenerically no instabilities. Black-hole disappearance in massive gravity was explored in Ref. [401].

Since all black-hole solutions of massive gravity arise as decoupling limits𝑀𝑓 →∞ of solutionsin bi-gravity,30 we can consider from the outset the bi-gravity solutions and consider the massivegravity limit after the fact. Let us consider then the bi-gravity action expressed as

𝑆 =𝑀2

Pl

2

∫d4𝑥√−𝑔𝑅[𝑔] +

𝑀2𝑓

2

∫d4𝑥√−𝑓𝑅[𝑓 ] (11.22)

+𝑚2𝑀2

eff

4

∫d4𝑥√−𝑔∑𝑛

𝛽𝑛𝑛!ℒ𝑛(√X) +Matter ,

where 𝑀−2eff = 𝑀−2

Pl + 𝑀−2𝑓 . Here, the definition is such that in the limit 𝑀𝑓 → ∞ the 𝛽𝑛’s

correspond the usual expressions in massive gravity. We may imagine matter coupled to bothmetrics although to take the massive gravity limit we should imagine black holes formed frommatter which exclusively couples to the 𝑔 metric.

One immediate consequence of working with bi-gravity is that since the 𝑔 metric is sourced bypolynomials of

√X =

√𝑔−1𝑓 whereas the 𝑓 metric is sourced by polynomials of

√𝑓−1𝑔. We, thus,

require that X is invertible away from curvature singularities. This is equivalent to saying thatthe eigenvalues of 𝑔−1𝑓 and 𝑓−1𝑔 should not pass through zero away from a curvature singularity.This in turn means that if one metric is diagonal and admits a horizon, the second metric if it isdiagonal must admit a horizon at the same place, i.e., two diagonal metrics have common horizons.This is a generic observation that is valid for any theory with more than one metric [167] regardlessof the field equations. Equivalently, this implies that if 𝑓 is a diagonal metric without horizons,e.g., Minkowski spacetime, then the metric for a black hole must be non-diagonal when workingin unitary gauge. This is consistent with the known exact solutions. For certain solutions it maybe possible by means of introducing Stuckelberg fields to put both metrics in diagonal form, due

30 In taking this limit, it is crucial that the second metric 𝑓𝜇𝜈 be written in a locally inertial coordinate system,i.e., a system which is locally Minkowski. Failure to do this will lead to the erroneous conclusion that massivegravity on Minkowski is not a limit of bi-gravity.



Massive Gravity 127

to the Stuckelberg fields absorbing the off-diagonal terms. However, for the generic solution wewould expect that at least one metric to be non-diagonal even with Stuckelberg fields present.

Working with a static spherically symmetric ansatz for both metrics, we find in general thatbi-gravity admits Schwarzschild-(anti) de Sitter-type metrics of the form (see [478] for a review)

d𝑠2𝑔 = −𝐷(𝑟) d𝑡2 +1

𝐷(𝑟)d𝑟2 + 𝑟2 dΩ2 , (11.23)

d𝑠2𝑓 = −Δ(𝑈) d𝑇 2 +1

Δ(𝑈)d𝑈2 + 𝑈2 dΩ2 , (11.24)

where

𝐷(𝑟) = 1− 2𝑀

8𝜋𝑀2Pl𝑟− 1

3Λ𝑔𝑟

2 , (11.25)

Δ(𝑈) = 1− 1

3Λ𝑓𝑈

2 , (11.26)

are the familiar metric functions for de Sitter and Schwarzschild–de Sitter.The 𝑓 -metric coordinates are related to those of the 𝑔 metric by (in other words the profiles of

the Stuckelberg fields)

𝑈 = 𝑢𝑟 , 𝑇 = 𝑢𝑡− 𝑢∫𝐷(𝑟)−Δ(𝑈)

𝐷(𝑟)Δ(𝑈)d𝑟 , (11.27)

where the constant 𝑢 is given by

𝑢 = −2𝛽2𝛽3± 1

𝛽3

√4𝛽2

2 − 6𝛽1𝛽3 . (11.28)

Finally, the two effective cosmological constants that arise from the mass terms are

Λ𝑔 = −𝑚2𝑀2

eff

𝑀2Pl

(6𝛽0 + 2𝛽1𝑢+

1

2𝛽2𝑢

2

), (11.29)

Λ𝑓 = −𝑚2𝑀2

eff

𝑀2𝑓𝑢

2

(1

2𝛽2 +

1

2𝛽3𝑢+

1

4𝛽4𝑢

2

). (11.30)

In this form, we see that in the limit 𝑀𝑓 → ∞ we have Λ𝑓 → 0 and 𝑀eff → 𝑀Pl and then thesesolutions match onto the known exact black holes solutions in massive gravity in the absence ofcharge [413, 363, 365, 277, 105, 56, 477, 478, 455, 30, 357]. Note in particular that for every set of𝛽𝑛’s there are two branches of solutions determined by the two possible values of 𝑢.

These solutions describe black holes sourced by matter minimally coupled to metric 𝑔 withmass 𝑀 . An obvious generalization is to assume that the matter couples to both metrics, witheffective masses 𝑀1 and 𝑀2 so that

𝐷(𝑟) = 1− 2𝑀1

8𝜋𝑀2Pl𝑟− 1

3Λ𝑔𝑟

2 , (11.31)

Δ(𝑈) = 1− 2𝑀2

8𝜋𝑀2𝑓𝑈− 1

3Λ𝑓𝑈

2 . (11.32)

Although these are exact solutions, not all of them are stable for all values and ranges of parametersand in certain cases it is found that the quadratic kinetic term for various fluctuations vanishesindicating a linearization instability, which means these are not good vacuum solutions. On theother hand, neither are these the most general black-hole–like solutions; the general case requires



128 Claudia de Rham

numerical analysis to solve the equations which is a subject of ongoing work (see, e.g., [477]). Wenote only that in [477] a distinct class of solutions is obtained numerically in bi-gravity for whichthe two metrics take the diagonal form

d𝑠2𝑔 = −𝑄(𝑟)2 d𝑡2 +1

𝑁(𝑟)2d𝑟2 + 𝑟2 dΩ2 , (11.33)

d𝑠2𝑓 = −𝐴(𝑟)2 d𝑇 2 +𝑈 ′(𝑟)2

𝑌 (𝑟)2d𝑟2 + 𝑈(𝑟)2 dΩ2 , (11.34)

where 𝑄,𝑁,𝐴, 𝑌, 𝑈 are five functions of radius that are numerically obtained solutions of fivedifferential equations. According to the previous arguments about diagonal metrics [167] thesesolutions do not correspond to black holes in the massive gravity on Minkowski limit 𝑀𝑓 → ∞,however the limit 𝑀𝑓 →∞ can be taken and they correspond to black-hole solutions in a theoryof massive gravity in which the reference metric is Schwarzschild (–de Sitter or anti-de Sitter). Thearguments of [167] are then evaded since the reference metric itself admits a horizon.

12 Cosmology

One of the principal motivations for considering massive theories of gravity is their potential toaddress, or at least provide a new perspective on, the issue of cosmic acceleration as alreadydiscussed in Section 3. Adding a mass for the graviton keeps physics at small scales largelyequivalent to GR because of the Vainshtein mechanism. However, it inevitably modifies gravityin at large distances, i.e., in the infrared. This modification of gravity is thus most significant forsources which are long wavelength. The cosmological constant is the most infrared source possiblesince it is build entirely out of zero momentum modes and for this reason we may hope that thenature of a cosmological constant in a theory of massive gravity or similar infrared modification ischanged.

There have been two principal ideas for how massive theories of gravity could be useful foraddressing the cosmological constant. On the one hand, by weakening gravity in the infrared, theymay weaken the sensitivity of the dynamics to an already existing large cosmological constant.This is the idea behind screening or degravitating solutions [211, 212, 26, 216] (see Section 4.5).The second idea is that a condensate of massive gravitons could form which act as a source for self-acceleration, potentially explaining the current cosmic acceleration without the need to introducea non-zero cosmological constant (as in the case of the DGP model [159, 163], see Section 4.4).This idea does not address the ‘old cosmological constant problem’ [484] but rather assumes thatsome other symmetry, or mechanism exists which ensures the vacuum energy vanishes. Given this,massive theories of gravity could potential provide an explanation for the currently small, andhence technically unnatural value of the cosmological constant, by tying it to the small, technicallynatural, value of the graviton mass.

Thus, the idea of screening/degravitation and self-acceleration are logically opposites to eachother, but there is some evidence that both can be achieved in massive theories of gravity. Thisevidence is provided by the decoupling limit of massive gravity to which we review first. We thengo on to discuss attempts to find exact solutions in massive gravity and its various extensions.

12.1 Cosmology in the decoupling limit

A great deal of understanding about the cosmological solutions in massive gravity theories can belearned from considering the ‘decoupling limit’ of massive gravity discussed in Section 8.3. Theidea here is to recognize that locally, i.e., in the vicinity of a point, any FLRW geometry can be



Massive Gravity 129

expressed as a small perturbation about Minkowski spacetime (about �� = 0) with the perturbationexpansion being good for distances small relative to the curvature radius of the geometry:

d𝑠2 = −[1− (�� +𝐻2)��2

]d𝑡2 +

[1− 1

2𝐻2��2

]d��2 (12.1)

=

(𝜂𝜇𝜈 +

1

𝑀PlℎFLRW𝜇𝜈

)d𝑥𝜇 d𝑥𝜈 . (12.2)

In the decoupling limit 𝑀Pl →∞, 𝑚→ 0 we keep the canonically normalized metric perturbationℎ𝜇𝜈 fixed. Thus, the decoupling limit corresponds to keeping 𝐻2𝑀Pl and ��𝑀Pl fixed, or equiva-

lently 𝐻2/𝑚2 and ��/𝑚2 fixed. Despite the fact that 𝐻 → 0 vanishes in this limit, the analogueof the Friedmann equation remains nontrivial if we also scale the energy density such that 𝜌/𝑀Pl

remains finite. Because of this fact, it is possible to analyze the modification to the Friedmannequation in the decoupling limit.31

The generic form for the helicity-0 mode which preserves isotropy near �� = 0 is

𝜋 = 𝐴(𝑡) +𝐵(𝑡)��2 + . . . . (12.3)

In the specific case where ��(𝑡) = 0, this also preserves homogeneity in a theory in which theGalileon symmetry is exact, as in massive gravity, since a translation in �� corresponds to a Galileontransformation of 𝜋 which leaves invariant the combination 𝜕𝜇𝜕𝜈𝜋. In Ref. [139] this ansatz wasused to derive the existence of both self-accelerating and screening solutions.

Friedmann equation in the decoupling limit

We start with the decoupling limit Lagrangian given in (8.38). Following the same notation as inRef. [139] we set 𝑎𝑛 = −𝑐𝑛/2, where the coefficients 𝑐𝑛 are given in terms of the 𝛼𝑛’s in (8.47).The self-accelerating branch of solutions then corresponds to the ansatz

𝜋 =1

2𝑞0Λ

33𝑥𝜇𝑥

𝜇 + 𝜑 (12.4)

ℎ𝜇𝜈 = −1

2𝐻2

dS𝑥𝜇𝑥𝜇 + 𝜒𝜇𝜈 (12.5)

𝑇𝜇𝜈 = −𝜆𝜂𝜇𝜈 + 𝜏𝜇𝜈 , (12.6)

where 𝜋, 𝜒 and 𝜏 correspond to the fluctuations about the background solution.

For this ansatz, the background equations of motion reduce to

𝐻dS

(𝑎1 + 2𝑎2𝑞0 + 3𝑎3𝑞

20

)= 0 (12.7)

𝐻2dS =

𝜆

3𝑀2Pl

+2Λ3

3

𝑀Pl(𝑎1𝑞0 + 𝑎2𝑞

20 + 𝑎3𝑞

30) . (12.8)

In the ‘self-accelerating branch’ when 𝐻dS = 0, the first constraint can be used to infer 𝑞0 andthe second one corresponds to the effective Friedmann equation. We see that even in the absenceof a cosmological constant 𝜆 = 0, for generic coefficients we have a constant 𝐻dS solution whichcorresponds to a self-accelerating de Sitter solution.

31 In the context of DGP, the Friedmann equation was derived in Section 4.3.1 from the full five-dimensionalpicture, but one would have obtained the correct result if derived instead from the decoupling limit. The reason isthe main modification of the Friedmann equation arises from the presence of the helicity-0 mode which is alreadycaptured in the decoupling limit.



130 Claudia de Rham

The stability of these solutions can be analyzed by looking at the Lagrangian for the quadraticfluctuations

ℒ(2) = −1

4𝜒𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝜒𝛼𝛽 +

6𝐻2dS𝑀Pl

Λ33

(𝑎2 + 3𝑎3𝑞0)𝜑2𝜑+1

𝑀Pl𝜒𝜇𝜈𝜏𝜇𝜈 . (12.9)

Thus, we see that the helicity zero mode is stable provided that

(𝑎2 + 3𝑎3𝑞0) > 0 . (12.10)

However, these solutions exhibit a peculiarity. To this order, the helicity-0 mode fluctuations donot couple to the matter perturbations (there is no kinetic mixing between 𝜋 and 𝜒𝜇𝜈). This meansthat there is no Vainshtein effect, but at the same time there is no vDVZ discontinuity for theVainshtein effect to resolve!

Screening solution

Another way to solve the system of Eqs. (12.7) and (12.8) is to consider instead flat solutions𝐻dS = 0. Then (12.7) is trivially satisfied and we see the existence of a ‘screening solution’in the Friedmann equation (12.8), which can accommodate a cosmological constant without anyacceleration. This occurs when the helicity-0 mode ‘absorbs’ the contribution from the cosmologicalconstant 𝜆, and the background configuration for 𝜋 parametrized by 𝑞0 satisfies

(𝑎1𝑞0 + 𝑎2𝑞20 + 𝑎3𝑞

30) = −

𝜆

6𝑀PlΛ33

. (12.11)

Perturbations about this screened configuration then behave as

ℒ(2) = −1

2𝜒𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝜒𝛼𝛽 +

3

2𝜑2𝜑+

1

𝑀Pl(𝜒𝜇𝜈 + 𝜂𝜇𝜈𝜑)𝜏𝜇𝜈 . (12.12)

In this case the perturbations are stable, and the Vainshtein mechanism is present which is nec-essary to resolve the vDVZ discontinuity. Furthermore, since the background contribution to themetric perturbation vanishes ℎ𝜇𝜈 = 0, they correspond to Minkowski solutions which are sourcedby a nonzero cosmological constant. In the case where 𝑎3 = 0, these solutions only exist if

𝜆 < 𝑀PlΛ333𝑎212𝑎2

. In the case where 𝑎3 = 0, there is no upper bound on the cosmological constantwhich can be screened via this mechanism.

In this branch of solution, the strong coupling scale for fluctuations on top of this configurationbecomes of the same order of magnitude as that of the screened cosmological constant. For a largecosmological constant the strong coupling scale becomes to large and the helicity-0 mode wouldthus not be sufficiently Vainshtein screened.

Thus, while these solutions seem to indicate positively that there are self-screening solutionswhich can accommodate a continuous range of values for the cosmological constant and still re-main flat, the range is too small to significantly change the old cosmological constant problem.Nevertheless, the considerable difficulty in attacking the old cosmological constant problem meansthat these solutions deserve further attention as they also provide a proof of principle on howWeinberg’s no go could be evaded [484]. We emphasize that what prevents a large cosmologicalconstant from being screened is not an issue in the theoretical tuning but rather an observationalbound, so this is already a step forward.

These two classes of solutions are both maximally symmetric. However, the general cosmo-logical solution is isotropic but inhomogeneous. This is due to the fact that a nontrivial timedependence for the matter source will inevitably source 𝐵(𝑡), and as soon as �� = 0 the solutionsare inhomogeneous. In fact, as we now explain in general, the full nonlinear solution is inevitablyinhomogeneous due to the existence of a no-go theorem against spatially flat and closed FLRWsolutions.



Massive Gravity 131

12.2 FLRW solutions in the full theory

12.2.1 Absence of flat/closed FLRW solutions

A nontrivial consequence of the fact that diffeomorphism invariance is broken in massive gravityis that there are no spatially flat or closed FLRW solutions [117]. This result follows from thedifferent nature of the Hamiltonian constraint. For instance, choosing a spatially flat form for themetric d𝑠2 = − d𝑡2 + 𝑎(𝑡)2 d��2, the mini-superspace Lagrangian takes the schematic form

ℒ = −3𝑀2Pl

𝑎��2

𝑁+ 𝐹1(𝑎) + 𝐹2(𝑎)𝑁 . (12.13)

Consistency of the constraint equation obtained from varying with respect to𝑁 and the accelerationequation for �� implies

𝜕𝐹1(𝑎)

𝜕𝑡= ��

𝜕𝐹1(𝑎)

𝜕𝑎= 0 . (12.14)

In GR, since 𝐹1(𝑎) = 0, there is no analogue of this equation. In the present case, this equationcan be solved either by imposing �� = 0 which implies the absence of any dynamic FLRW solutions,

or by solving 𝜕𝐹1(𝑎)𝜕𝑎 = 0 for fixed 𝑎 which implies the same thing. Thus, there are no nontrivial

spatially flat FLRW solutions in massive gravity in which the reference metric is Minkowski. Theresult extends also to spatially closed cosmological solutions. As a result, different alternativeshave been explored in the literature to study the cosmology of massive gravity. See Figure 7 for asummary of these different approaches.

12.2.2 Open FLRW solutions

While the previous argument rules out the possibility of spatially flat and closed FLRW solutions,open ones are allowed [283]. To see this we make the ansatz d𝑠2 = − d𝑡2 + 𝑎(𝑡)2 dΩ2

𝐻3 , wheredΩ2

𝐻3 expressed in the form

dΩ2𝐻3 = d��2 − |𝑘| (��. d��)2

(1 + |𝑘|��2)=

d𝑟2

1 + |𝑘|𝑟2+ 𝑟2 dΩ2

𝑆2 , (12.15)

is the metric on a hyperbolic space, and express the reference metric in terms of Stuckelberg fields𝑓𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = 𝜂𝑎𝑏𝜕𝜇𝜑

𝑎𝜕𝜈𝜑𝑏 with

𝜑0 = 𝑓(𝑡)√1 + |𝑘|��2 , (12.16)

𝜑𝑖 =√|𝑘|𝑓(𝑡)𝑥𝑖 . (12.17)

then the mini-superspace Lagrangian of (6.3) takes the form

ℒmGR = −3𝑀2Pl|𝑘|𝑁𝑎− 3𝑀2

Pl

𝑎��2

𝑁+ 3𝑚2𝑀2

Pl

[2𝑎2𝒳 (2𝑁𝑎− 𝑓𝑎−𝑁

√|𝑘|𝑓

+ 𝛼3𝑎2𝒳 2(4𝑁𝑎− 3𝑓𝑎−𝑁

√|𝑘|𝑓) + 4𝛼4𝑎

3𝒳 3(𝑁 − 𝑓)],

with 𝒳 = 1 −√

|𝑘|𝑓𝑎 . In this case, the analogue additional constraint imposed by consistency of

the Friedmann and acceleration (Raychaudhuri) equation is

𝑎𝒳

((3−

2√|𝑘|𝑓𝑎

)+

3

2𝛼3

(3−

√|𝑘|𝑓𝑎

)𝒳 + 6𝛼4𝒳 2

)= 0 . (12.18)



132 Claudia de Rham

No spatially-flat

FLRW solutions

with Minkowski

Reference metric

Large Scale Inhomogeneities

Open solutions (unstable)

dS or FRLW reference metric (problem with Higuchi ghost)

Additional degrees of freedom

Scalar Tensor

Quasi-dilaton

Extended Quasi-dilaton

Mass-Varying

f(R)

…

Bi-Gravity

Multi-Gravity

Figure 7: Alternative ways in deriving the cosmology in massive gravity.

The solution for which 𝒳 = 0 is essentially Minkowski spacetime in the open slicing, and is thusuninteresting as a cosmology.

Focusing on the other branch and assuming 𝒳 = 0, the general solution determines 𝑓(𝑡) interms of 𝑎(𝑡) takes the form 𝑓(𝑡) = 1√

|𝑘|𝑢𝑎(𝑡) where 𝑢 is a constant determined by the quadratic

equation

3− 2𝑢+3

2𝛼3 (3− 𝑢) (1− 𝑢) + 6𝛼4(1− 𝑢)2 = 0 . (12.19)

The resulting Friedmann equation is then

3𝑀2Pl𝐻

2 − 3𝑀2Pl|𝑘|𝑎2

= 𝜌+ 2𝑚2𝜌𝑚 , (12.20)

where

𝜌𝑚 = −(1− 𝑢)(3 (2− 𝑢) + 3

2𝛼3 (4− 𝑢) (1− 𝑢) + 6𝛼4(1− 𝑢)2

). (12.21)

Despite the positive existence of open FLRW solutions in massive gravity, there remain problemsof either strong coupling (due to absence of quadratic kinetic terms for physical degrees of freedom)or other instabilities which essentially rule out the physical relevance of these FLRW solutions [285,125, 464].



Massive Gravity 133

12.3 Inhomogenous/anisotropic cosmological solutions

As pointed out in [117], the absence of FLRW solutions in massive gravity should not be viewed asan observational flaw of the theory. On the contrary, the Vainshtein mechanism guarantees thatthere exist inhomogeneous cosmological solutions which approximate the normal FLRW solutionsof GR as closely as desired in the limit 𝑚→ 0. Rather, it is the existence of a new physical lengthscale 1/𝑚 in massive gravity, which cause the dynamics to be inhomogeneous at cosmologicalscales. If this scale 1/𝑚 is comparable to or larger than the current Hubble radius, then the effectsof these inhomogeneities would only become apparent today, with the universe locally appearingas homogeneous for most of its history in the local patch that we observe.

One way to understand how the Vainshtein mechanism recovers the prediction of homogeneityand isotropy is to work in the formulation of massive gravity in which the Stuckelberg fields areturned on. In this formulation, the Stuckelberg fields can exhibit order unity inhomogeneities withthe metric remaining approximately homogeneous. Matter that couples only to the metric willperceive an effectively homogeneous and anisotropic universe, and only through interaction withthe Vainshtein suppressed additional scalar and vector degrees of freedom would it be possible toperceive the inhomogeneities. This is achieved because the metric is sourced by the Stuckelbergfields through terms in the equations of motion which are suppressed by 𝑚2. Thus, as long as𝑅 ≫ 𝑚2, the metric remains effectively homogeneous and isotropic despite the existence of no-gotheorems against exact homogeneity and isotropy.

In this regard, a whole range of exact solutions have been studied exhibiting these proper-ties [364, 474, 363, 97, 264, 356, 456, 491, 476, 334, 478, 265, 124, 123, 125, 455, 198]. A general-ization of some of these solutions was presented in Ref. [404] and Ref. [266]. In particular, we notethat in [475, 476] the most general exact solution of massive gravity is obtained in which the met-ric is homogeneous and isotropic with the Stuckelberg fields inhomogeneous. These solutions existbecause the effective contribution to the stress energy tensor from the mass term (i.e., viewing themass term corrections as a modification to the energy density) remains homogeneous and isotropicdespite the fact that it is build out of Stuckelberg fields which are themselves inhomogeneous.

Let us briefly discuss how these solutions are obtained.32 As we have already discussed, allsolutions of massive gravity can be seen as 𝑀𝑓 →∞ decoupling limits of bi-gravity. Therefore, wemay consider the case of inhomogeneous solutions in bi-gravity and the solutions of massive gravitycan always be derived as a limit of these bi-gravity solutions. We thus begin with the action

𝑆 =𝑀2

Pl

2

∫d4𝑥√−𝑔𝑅[𝑔] +

𝑀2𝑓

2

∫d4𝑥√−𝑓𝑅[𝑓 ] (12.22)

+𝑚2𝑀2

eff

4

∫d4𝑥√−𝑔∑𝑛

𝛽𝑛𝑛!ℒ𝑛(√X) +Matter ,

where 𝑀−2eff = 𝑀−2

Pl +𝑀−2𝑓 and

√X =

√𝑔−1𝑓 and we may imagine matter coupled to both 𝑓

and 𝑔 but for simplicity let us imagine matter is either minimally coupled to 𝑔 or it is minimallycoupled to 𝑓 .

12.3.1 Special isotropic and inhomogeneous solutions

Although it is possible to find solutions in which the two metrics are proportional to each other𝑓𝜇𝜈 = 𝐶2𝑔𝜇𝜈 [478], these solutions require in addition that the stress energies of matter sourcing 𝑓and 𝑔 are proportional to one another. This is clearly too restrictive a condition to be phenomeno-logically interesting. A more general and physically realistic assumption is to suppose that both

32 See [478] for a recent review and more details. The convention on the parameters 𝑏𝑛 there is related to our𝛽𝑛’s here via 𝑏𝑛 = − 1

4(4− 𝑛)!𝛽𝑛.



134 Claudia de Rham

metrics are isotropic but not necessarily homogeneous. This is covered by the ansatz

d𝑠2𝑔 = −𝑄(𝑡, 𝑟)2 d𝑡2 +𝑁(𝑡, 𝑟)2 d𝑟2 +𝑅2 d2Ω𝑆2 , (12.23)

d𝑠2𝑓 = −(𝑎(𝑡, 𝑟)𝑄(𝑡, 𝑟)2 d𝑡+ 𝑐(𝑡, 𝑟)𝑁(𝑡, 𝑟) d𝑟)2 (12.24)

+(𝑐(𝑡, 𝑟)𝑄(𝑡, 𝑟) d𝑡− 𝑏(𝑡, 𝑟)𝑛(𝑡, 𝑟)𝑁(𝑡, 𝑟) d𝑟)2 + 𝑢(𝑡, 𝑟)2𝑅2 d2Ω𝑆2 ,

and d2Ω𝑆2 is the metric on a unit 2-sphere. To put the 𝑔 metric in diagonal form we have made useof the one copy of overall diff invariance present in bi-gravity. To distinguish from the bi-diagonalcase we shall assume that 𝑐(𝑡, 𝑟) = 0. The bi-diagonal case allows for homogeneous and isotropicsolutions for both metrics which will be dealt with in Section 12.4.2. The square root may be easilytaken to give

√X =

⎛⎜⎜⎝𝑎 𝑐𝑁/𝑄 0 0

−𝑐𝑁/𝑄 𝑏 0 00 0 𝑢 00 0 0 𝑢

⎞⎟⎟⎠ , (12.25)

which can easily be used to determine the contribution of the mass terms to the equations ofmotion for 𝑓 and 𝑔. This leads to a set of partial differential equations for 𝑄,𝑅,𝑁, 𝑛, 𝑐, 𝑏 whichin general require numerical analysis. As in GR, due to the presence of constraints associatedwith diffeomorphism invariance, and the Hamiltonian constraint for the massive graviton, severalof these equations will be first order in time-derivatives. This simplifies matters somewhat but notsufficiently to make analytic progress. Analytic progress can be made however by making addi-tional more restrictive assumptions, at the cost of potentially losing the most physically interestingsolutions.

Effective cosmological constant

For instance, from the above form we may determine that the effective contribution to the stressenergy tensor sourcing 𝑔 arising from the mass term is of the form

𝑇mass0𝑟 = −𝑚2𝑀

2eff

𝑀2Pl

𝑐𝑁

𝑄

(3

2𝛽1 + 𝛽2𝑢+

1

4𝛽3𝑢

2

). (12.26)

If we make the admittedly restrictive assumption that the metric 𝑔 is of the FLRW form or isstatic, then this requires that 𝑇 0

𝑟=0 which for 𝑐 = 0 implies

3

2𝛽1 + 𝛽2𝑢+

1

4𝛽3𝑢

2 = 0 . (12.27)

This should be viewed as an equation for 𝑢(𝑡, 𝑟) whose solution is

𝑢(𝑡, 𝑟) = 𝑢 = −2𝛽2𝛽3± 1

𝛽3

√4𝛽2

2 − 6𝛽1𝛽3 . (12.28)

Then conservation of energy imposes further

𝑇mass00 − 𝑇mass

𝜃𝜃=−𝑚2𝑀

2eff

𝑀2Pl

(1

2𝛽2 +

1

4𝛽3𝑢

)((𝑢− 𝑎)(𝑢− 𝑏) + 𝑐2

)= 0 , (12.29)

since 𝑢 is already fixed we should view this generically as an equation for 𝑐(𝑡, 𝑟) in terms of 𝑎(𝑡, 𝑟)and 𝑏(𝑡, 𝑟)

(𝑢− 𝑎)(𝑢− 𝑏) + 𝑐2 = 0 . (12.30)



Massive Gravity 135

With these assumptions the contribution of the mass term to the effective stress energy tensorsourcing each metric becomes equivalent to a cosmological constant for each metric 𝑇mass

𝜇𝜈(𝑔) =

−Λ𝑔𝛿𝜇𝜈 and 𝑇mass𝜇𝜈(𝑓) = −Λ𝑓𝛿𝜇𝜈 with

Λ𝑔 = −𝑚2𝑀2

eff

𝑀2Pl

(6𝛽0 + 2𝛽1𝑢+

1

2𝛽2𝑢

2

), (12.31)

Λ𝑓 = −𝑚2𝑀2

eff

𝑀2𝑓𝑢

2

(1

2𝛽2 +

1

2𝛽3𝑢+

1

4𝛽4𝑢

2

). (12.32)

Thus, all of the potential dynamics of the mass term is reduced to an effective cosmological constant.Let us stress again that this rather special fact is dependent on the rather restrictive assumptionsimposed on the metric 𝑔 and that we certainly do not expect this to be the case for the mostgeneral time-dependent, isotropic, inhomogeneous solution.

Massive gravity limit

As usual, we can take the 𝑀𝑓 → 0 limit to recover solutions for massive gravity on Minkowski(if Λ𝑓 → 0) or more generally if the scaling of the parameters 𝛽𝑛 is chosen so that Λ𝑓 and(6𝛽0 + 2𝛽1𝑢+ 1

2𝛽2𝑢2)and hence Λ𝑔 remains finite in the limit then these will give rise to solutions

for massive gravity for which the reference metric is any Einstein space for which

𝐺𝜇𝜈(𝑓) = −Λ𝑓𝑓𝜇𝜈 . (12.33)

For example, this includes the interesting cases of de Sitter and anti-de Sitter reference metrics.Thus, for example, assuming no additional matter couples to the 𝑓 metric, both bi-gravity and

massive gravity on a fixed reference metric admit exact cosmological solutions for which the 𝑓metric is de Sitter or anti-de Sitter

d𝑠2𝑔 = − d𝑡2 + 𝑎(𝑡)2(

d𝑟2

1− 𝑘𝑟2+ 𝑟2 dΩ2

)(12.34)

d𝑠2𝑓 = −Δ(𝑈) d𝑇 2 +

(d𝑈2

Δ(𝑈)+ 𝑈2 dΩ2

), (12.35)

where Δ(𝑈) = 1− Λ𝑓

3 𝑈2, and the scale factor 𝑎(𝑡) satisfies

3𝑀2Pl

(𝐻2 +

𝑘

𝑎2

)= Λ𝑔 + 𝜌𝑀 (𝑡) , (12.36)

where 𝜌𝑀 (𝑡) is the energy density of matter minimally coupled to 𝑔, 𝐻 = ��/𝑎, and 𝑈, 𝑇 canbe expressed as a function of 𝑟 and 𝑡 and comparing with the previous representation 𝑈 = 𝑢𝑟.The one remaining undetermined function is 𝑇 (𝑡, 𝑟) and this is determined by the constraint that(𝑢− 𝑎)(𝑢− 𝑏) + 𝑐2 = 0 and the conversion relations√

Δ(𝑈) d𝑇 = 𝑎(𝑡, 𝑟) d𝑡+ 𝑐(𝑡, 𝑟)1√

1− 𝑘𝑟2d𝑟 , (12.37)

1√Δ(𝑈)

d𝑈 = 𝑐(𝑡, 𝑟) d𝑡− 𝑏(𝑡, 𝑟)𝑛(𝑡, 𝑟) 1√1− 𝑘𝑟2

d𝑟 , (12.38)

𝑈(𝑡, 𝑟) = 𝑢𝑟 , (12.39)

which determine 𝑏(𝑡, 𝑟) and 𝑐(𝑡, 𝑟) in terms of �� and 𝑇 ′. These relations are difficult to solveexactly, but if we consider the special case Λ𝑓 = 0 which corresponds in particular to massive



136 Claudia de Rham

gravity on Minkowski then the solution is

𝑇 (𝑡, 𝑟) = 𝑞

∫ 𝑡

d𝑡1

��+

(𝑢2

4𝑞+ 𝑞𝑟2

)𝑎 , (12.40)

where 𝑞 is an integration constant.In particular, in the open universe case Λ𝑓 = 0, 𝑘 = 0, 𝑞 = 𝑢, 𝑇 = 𝑢𝑎

√1 + |𝑘|𝑟2, we recover

the open universe solution of massive gravity considered in Section 12.2.2, where for comparison𝑓(𝑡) = 1√

|𝑘|𝑢𝑎(𝑡), 𝜑0(𝑡, 𝑟) = 𝑇 (𝑡, 𝑟) and 𝜑𝑟 = 𝑈(𝑡, 𝑟).

12.3.2 General anisotropic and inhomogeneous solutions

Let us reiterate again that there are a large class of inhomogeneous but isotropic cosmologicalsolutions for which the effective Friedmann equation for the 𝑔 metric is the same as in GR withjust the addition of a cosmological constant which depends on the graviton mass parameters.However, these are not the most general solutions, and as we have already discussed many of theexact solutions of this form considered so far have been found to be unstable, in particular throughthe absence of kinetic terms for degrees of freedom which implies infinite strong coupling. However,all the exact solutions arise from making a strong restriction on one or the other of the metricswhich is not expected to be the case in general. Thus, the search for the ‘correct’ cosmologicalsolution of massive gravity and bi-gravity will almost certainly require a numerical solution of thegeneral equations for 𝑄,𝑅,𝑁, 𝑛, 𝑐, 𝑏, and their stability.

Closely related to this, we may consider solutions which maintain homogeneity, but are an-isotropic [284, 393, 123]. In [393] the general Bianchi class A cosmological solutions in bi-gravityare studied. There it is shown that the generic anisotropic cosmological solution in bi-gravityasymptotes to a self-accelerating solution, with an acceleration determined by the mass terms, butwith an anisotropy that falls off less rapidly than in GR. In particular the anisotropic contributionto the effective energy density redshifts like non-relativistic matter. In [284, 123] it is found thatif the reference metric is made to be of an anisotropic FLRW form, then for a range of parametersand initial conditions stable ghost free cosmological solutions can be found.

These analyses are ongoing and it has been uncovered that certain classes of exact solutions ex-hibit strong coupling instabilities due to vanishing kinetic terms and related pathologies. However,this simply indicates that these solutions are not good semi-classical backgrounds. The generalinhomogeneous cosmological solution (for which the metric is also inhomogeneous) is not known atpresent, and it is unlikely it will be possible to obtain it exactly. Thus, it is at present unclear whatare the precise nonlinear completions of the stable inhomogeneous cosmological solutions that canbe found in the decoupling limit. Thus the understanding of the cosmology of massive gravityshould be regarded as very much work in progress, at present it is unclear what semi-classicalsolutions of massive gravity are the most relevant for connecting with our observed cosmologicalevolution.

12.4 Massive gravity on FLRW and bi-gravity

12.4.1 FLRW reference metric

One straightforward extension of the massive gravity framework is to allow for modifications to thereference metric, either by making it cosmological or by extending to bi-gravity (or multi-gravity).In the former case, the no-go theorem is immediately avoided since if the reference metric is itselfan FLRW geometry, there can no longer be any obstruction to finding FLRW geometries.

The case of massive gravity with a spatially flat FLRW reference metric was worked out in [223],where it was found that if using the convention for which the massive gravity Lagrangian is (6.5)



Massive Gravity 137

with the potential given in terms of the coefficient 𝛽′𝑠 as in (6.23), then the Friedmann equationtakes the form

𝐻2 = −3∑

𝑛=0

3(4− 𝑛)𝑚2𝛽𝑛2𝑛!

(𝑏

𝑎

)𝑛+

1

3𝑀2Pl

𝜌 . (12.41)

Here the dynamical and reference metrics in the form

𝑔𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = − d𝑡2 + 𝑎(𝑡)2 d��2 , (12.42)

𝑓𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −𝑀2 d𝑡2 + 𝑏(𝑡)2 d��2 (12.43)

and the Hubble constants are related by

2∑𝑛=0

(3− 𝑛)𝛽𝑛+1

2𝑛!

(𝑏

𝑎

)𝑛+1(𝐻

𝑏− 𝐻𝑓

𝑎

)= 0 . (12.44)

Ensuring a nonzero ghost-free kinetic term in the vector sector requires us to always solve thisequation with

𝐻

𝐻𝑓=𝑏

𝑎, (12.45)

so that the Friedmann equation takes the form

𝐻2 = −3∑

𝑛=0

3(4− 𝑛)𝑚2𝛽𝑛2𝑛!

(𝐻

𝐻𝑓

)𝑛+

1

3𝑀2Pl

𝜌 , (12.46)

where 𝐻𝑓 is the Hubble parameter for the reference metric. By itself, this Friedmann equationlooks healthy in the sense that it admits FLRW solutions that can be made as close as desired tothe usual solutions of GR.

However, in practice, the generalization of the Higuchi consideration [307] to this case leads toan unacceptable bound (see Section 8.3.6).

It is a straightforward consequence of the representation theory for the de Sitter group that aunitary massive spin-2 representation only exists in four dimensions for 𝑚2 ≥ 2𝐻2 as was the casein de Sitter. Although this result only holds for linearized fluctuations around de Sitter, its originas a bound comes from the requirement that the kinetic term for the helicity zero mode is positive,i.e., the absence of ghosts in the scalar perturbations sector. In particular, the kinetic term for thehelicity-0 mode 𝜋 takes the form

ℒhelicity−0 ∝ −𝑚2(𝑚2 − 2𝐻2)(𝜕𝜋)2 . (12.47)

Thus, there should exist an appropriate generalization of this bound for any cosmological solutionof nonlinear massive gravity for which there an FLRW reference metric.

This generalized bound was worked out in [223] and takes the form

− 𝑚2

4𝑀2Pl

𝐻

𝐻𝑓

[3𝛽1 + 4𝛽2

𝐻

𝐻𝑓+ 𝛽3

𝐻2

𝐻2𝑓

]≥ 2𝐻2 . (12.48)

Again, by itself this equation is easy to satisfy. However, combined with the Friedmann equation,we see that the two equations are generically in conflict if in addition we require that the massivegravity corrections to the Friedmann equation are small for most of the history of the Universe,i.e., during radiation and matter domination

−3∑

𝑛=0

3(4− 𝑛)𝑚2𝛽𝑛2𝑛!

(𝐻

𝐻𝑓

)𝑛≤ 𝐻2 . (12.49)



138 Claudia de Rham

This phenomenological requirement essentially rules out the applicability of FLRW cosmologicalsolutions in massive gravity with an FLRW reference metric.

This latter problem which is severe for massive gravity with dS or FLRW reference metrics,33

gets resolved in bi-gravity extensions, at least for a finite regime of parameters.

12.4.2 Bi-gravity

Cosmological solutions in bi-gravity have been considered in [474, 479, 104, 106, 8, 475, 478, 9, 7, 62].We keep the same notation as previously and consider the action for bi-gravity as in (5.43) (interms of the 𝛽’s where the conversion between the 𝛽’s and the 𝛼’s is given in (6.28))

ℒbi−gravity =𝑀2

Pl

2

√−𝑔𝑅[𝑔] +

𝑀2𝑓

2

√−𝑓𝑅[𝑓 ]

+𝑚2𝑀2

Pl

4

4∑𝑛=0

𝛽𝑛𝑛!ℒ𝑛[√X] + ℒmatter[𝑔, 𝜓𝑖] , (12.50)

assuming that matter only couples to the 𝑔 metric. Then the two Friedmann equations for eachHubble parameter take the respective form

𝐻2 = −3∑

𝑛=0

3(4− 𝑛)𝑚2𝛽𝑛2𝑛!

(𝐻

𝐻𝑓

)𝑛+

1

3𝑀2Pl

𝜌 (12.51)

𝐻2𝑓 = −𝑀

2Pl

𝑀2𝑓

[3∑

𝑛=0

3𝑚2𝛽𝑛+1

2𝑛!

(𝐻

𝐻𝑓

)𝑛−3]. (12.52)

Crucially, the generalization of the Higuchi bound now becomes

−𝑚2

4

𝐻

𝐻𝑓

[3𝛽1 + 4𝛽2

𝐻

𝐻𝑓+ 𝛽3

𝐻2

𝐻2𝑓

][1 +

(𝐻𝑓𝑀Pl

𝐻𝑀𝑓

)2]≥ 2𝐻2 . (12.53)

The important new feature is the last term in square brackets. Although this tends to unity inthe limit 𝑀𝑓 → ∞, which is consistent with the massive gravity result, for finite 𝑀𝑓 it opens a

new regime where the bound is satisfied by having(𝐻𝑓𝑀Pl

𝐻𝑀𝑓

)2≫ 1 (notice that in our convention

the 𝛽’s are typically negative). One may show [224] that it is straightforward to find solutions ofboth Friedmann equations which are consistent with the Higuchi bound over the entire history ofthe universe. For example, choosing the parameters 𝛽2 = 𝛽3 = 0 and solving for 𝐻𝑓 the effectiveFriedmann equation for the metric which matter couples to is

𝐻2 =1

6𝑀2Pl

(𝜌(𝑎) +

√𝜌(𝑎)2 +

12𝑚4𝑀6Pl

𝑀2𝑓

)(12.54)

and the generalization of the Higuchi bound is(1 +

16𝑀2𝑓

3𝑀2Pl𝛽

21

𝐻4

𝑚4

)> 0 . (12.55)

which is trivially satisfied at all times. More generally, there is an open set of such solutions.The observationally viability of the self-accelerating branch of these models has been considered

33 Notice that this is not an issue in massive gravity with a flat reference metric since the analogue Friedmannequation does not even exist.



Massive Gravity 139

in [8, 9] with generally positive results. Growth histories of the bi-gravity cosmological solutionshave been considered in [62]. However, while avoiding the Higuchi bound indicates absence ofghosts, it has been argued that these solutions may admit gradient instabilities in their cosmologicalperturbations [106].

We should stress again that just as in massive gravity, the absence of FLRW solutions shouldnot be viewed as an inconsistency of the theory with observations, also in bi-gravity these solutionsmay not necessarily be the ones of most relevance for connecting with observations. It is only thatthey are the most straightforward to obtain analytically. Thus, cosmological solutions in bi-gravity,just as in massive gravity, should very much be viewed as a work in progress.

12.5 Other proposals for cosmological solutions

Finally, we may note that more serious modifications the massive gravity framework have beenconsidered in order to allow for FLRW solutions. These include mass-varying gravity and thequasi-dilaton models [119, 118]. In [281] it was shown that mass-varying gravity and the quasi-dilaton model could allow for stable cosmological solutions but for the original quasi-dilaton theorythe self-accelerating solutions are always unstable. On the other hand, the generalizations of thequasi-dilaton [126, 127] appears to allow stable cosmological solutions.

In addition, one can find cosmological solutions in non-Lorentz invariant versions of mas-sive gravity [109] (and [103, 107, 108]). We can also allow the mass to become dependent ona field [489, 375], extend to multiple metrics/vierbeins [454], extensions with 𝑓(𝑅) terms either inmassive gravity [89] or in bi-gravity [416, 415] which leads to interesting self-accelerating solutions.Alternatively, one can consider other extensions to the form of the mass terms by coupling massivegravity to the DBI Galileons [237, 19, 20, 315].

As an example, we present here the cosmology of the extension of the quasi-dilaton modelconsidered in [127], where the reference metric 𝑓𝜇𝜈 is given in (9.15) and depends explicitly on thedynamical quasi-dilaton field 𝜎.

The action takes the familiar form with an additional kinetic term introduced for the quasi-dilaton which respects the global symmetry

𝑆 =𝑀2Pl

∫d4𝑥√−𝑔[1

2𝑅− Λ− 𝜔

2𝑀2Pl

(𝜕𝜎)2

+𝑚2

4

(ℒ2[��] + 𝛼3ℒ3[��] + 𝛼4ℒ4[��]

)], (12.56)

where the tensor �� is given in (9.14).The background ansatz is taken as

d𝑠2 = −𝑁2(𝑡) d𝑡2 + 𝑎(𝑡)2 d��2 , (12.57)

𝜑0 = 𝜑0(𝑡) , (12.58)

𝜑𝑖 = 𝑥𝑖 , (12.59)

𝜎 = 𝜎(𝑡) , (12.60)

so that

𝑓𝜇𝜈 d𝑥𝜇 d𝑥𝜈 = −𝑛(𝑡)2 d𝑡2 + d��2 , (12.61)

𝑛(𝑡)2 = (��0(𝑡))2 +𝛼𝜎

𝑀2Pl𝑚

2��2 . (12.62)

The equation that for normal massive gravity forbids FLRW solutions follows from varying withrespect to 𝜑0 and takes the form

𝜕𝑡(𝑎4𝑋(1−𝑋)𝐽

)= 0 , (12.63)



140 Claudia de Rham

where

𝑋 =𝑒𝜎/𝑀Pl

𝑎and 𝐽 = 3 +

9

2(1−𝑋)𝛼3 + 6(1−𝑋)2𝛼4 . (12.64)

As the universe expands 𝑋(1 − 𝑋)𝐽 ∼ 1/𝑎4 which for one branch of solutions implies 𝐽 → 0which determines a fixed constant asymptotic value of 𝑋 from 𝐽 = 0. In this asymptotic limit theeffective Friedmann equation becomes(

3

2− 𝜔

)𝐻2 = Λ+ Λ𝑋 , (12.65)

where

Λ𝑋 = 𝑚2(𝑋 − 1)

[6− 3𝑋 +

3

2(𝑋 − 4)(𝑋 − 1)𝛼3 + 6(𝑋 − 1)2𝛼4

], (12.66)

defines an effective cosmological constant which gives rise to self-acceleration even when Λ = 0 (for𝜔 < 6).

The analysis of [127] shows that these self-accelerating cosmological solutions are ghost freeprovided that

0 < 𝜔 < 6 , 𝑋2 <𝛼𝜎𝐻

2

𝑚2𝑔

< 𝑟2𝑋2 (12.67)

where

𝑟 = 1 +𝜔𝐻2

𝑚2𝑋2( 32𝛼3(𝑋 − 1)− 2). (12.68)

In particular, this implies that 𝛼𝜎 > 0 which demonstrates that the original quasi-dilaton model [119,116] has a scalar (Higuchi type) ghost. The analysis of [126] confirms these properties in a moregeneral extension of this model.



Massive Gravity 141

Part IV

Other Theories of Massive Gravity

13 New Massive Gravity

13.1 Formulation

Independently of the formal development of massive gravity in four dimensions described above,there has been interest in constructing a purely three dimensional theory of massive gravity. Threedimensions are special for the following reason: for a massless graviton in three dimensions there areno propagating degrees of freedom. This follows simply by counting, a symmetric tensor in threedimensions has six components. A massless graviton must admit a diffeomorphism symmetry whichrenders three of the degrees of freedom pure gauge, and the remaining three are non-dynamical dueto the associated first class constraints. On the contrary, a massive graviton in three dimensionshas the same number of degrees of freedom as a massless graviton in four dimensions, namelytwo. Combining these two facts together, in three dimensions it should be possible to constructa diffeomorphism invariant theory of massive gravity. The usual massless graviton implied bydiffeomorphism invariance is absent and only the massive degree of freedom remains.

A diffeomorphism and parity invariant theory in three dimensions was given in [66] and referredto as ‘new massive gravity’ (NMG). In its original formulation the action is taken to be

𝑆NMG =1

𝜅2

∫d3𝑥√−𝑔[𝜎𝑅+

1

𝑚2

(𝑅𝜇𝜈𝑅

𝜇𝜈 − 3

8𝑅2

)], (13.1)

where 𝜅2 = 1/𝑀3 defines the three dimensional Planck mass, 𝜎 = ±1 and 𝑚 is the mass of thegraviton. In this form the action is manifestly diffeomorphism invariant and constructed entirelyout of the metric 𝑔𝜇𝜈 . However, to see that it really describes a massive graviton, it is helpfulto introduce an auxiliary field 𝑓𝜇𝜈 which, as we will see below, also admits an interpretation as ametric, to give a quasi-bi-gravity formulation

𝑆NMG =𝑀3

∫d3𝑥√−𝑔[𝜎𝑅− 𝑞𝜇𝜈𝐺𝜇𝜈 −

1

4𝑚2(𝑞𝜇𝜈𝑞

𝜇𝜈 − 𝑞2)]. (13.2)

The kinetic term for 𝑞𝜇𝜈 appears from the mixing with 𝐺𝜇𝜈 . Although this is not a true bi-gravitytheory, since there is no direct Einstein–Hilbert term for 𝑞𝜇𝜈 , we shall see below that it is a well-defined decoupling limit of a bi-gravity theory, and for this reason it makes sense to think of 𝑞𝜇𝜈 aseffectively a metric degree of freedom. In this form, we see that the special form of 𝑅2

𝜇𝜈 − 3/8𝑅2

was designed so that 𝑞𝜇𝜈 has the Fierz–Pauli mass term. It is now straightforward to see that thiscorresponds to a theory of massive gravity by perturbing around Minkowski spacetime. Defining

𝑔𝜇𝜈 = 𝜂𝜇𝜈 +1√𝑀3

ℎ𝜇𝜈 , (13.3)

and perturbing to quadratic order in ℎ𝜇𝜈 and 𝑞𝜇𝜈 we have

𝑆2 =𝑀3

∫d3𝑥

[−𝜎2ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 − 𝑞𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 −

1


𝜇𝜈 − 𝑞2)]. (13.4)

Finally, diagonalizing as ℎ𝜇𝜈 = ℎ𝜇𝜈 − 𝜎𝑞𝜇𝜈 we obtain

𝑆2 =𝑀3

∫d3𝑥

[−𝜎2ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 +

𝜎

2𝑞𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝑞𝛼𝛽 −

1


𝜇𝜈 − 𝑞2)], (13.5)



142 Claudia de Rham

which is manifestly a decoupled massless graviton and massive graviton. Crucially, however, wesee that the kinetic terms of each have the opposite sign. Since only the degrees of freedom of themassive graviton 𝑞𝜇𝜈 are propagating, unitarity when coupled to other sources forces us to choose𝜎 = −1. The apparently ghostly massless graviton does not lead to any unitarity violation, atleast in perturbation theory, as there is no massless pole in the propagator. The stability of thevacua was further shown in different gauges in Ref. [252].

13.2 Absence of Boulware–Deser ghost

The auxiliary field formulation of new massive gravity is also useful for understanding the absence ofthe BD ghost [141]. Setting 𝜎 = −1 as imposed previously and working with the formulation (13.2),we can introduce new vector and scalar degrees of freedom as follows

𝑞𝜇𝜈 =1√𝑀3

𝑞𝜇𝜈 +∇𝜇𝑉𝜈 +∇𝜈𝑉𝜇 , (13.6)

with

𝑉𝜇 =1√𝑀3𝑚

𝐴𝜇 +∇𝜇𝜋√𝑀3𝑚2

, (13.7)

where the factors of√𝑀3 and𝑚 are chosen for canonical normalization. 𝐴𝜇 represents the helicity-

1 mode which carries 1 degree of freedom and 𝜋 the helicity-0 mode that carries 1 degree freedom.These two modes carries all the dynamical fields.

Introducing new fields in this way also introduced new symmetries. Specifically there is a 𝑈(1)symmetry

𝜋 → 𝜋 +𝑚𝜒 , 𝐴𝜇 → 𝐴𝜇 − 𝜒 , (13.8)

and a linear diffeomorphism symmetry

𝑞𝜇𝜈 → 𝑞𝜇𝜈 +∇𝜇𝜒𝜈 +∇𝜈𝜒𝜇 , 𝐴𝜇 → 𝐴𝜇 −√𝑚𝜒𝜇 . (13.9)

Substituting in the action, integrating by parts and using the Bianchi identity ∇𝜇𝐺𝜇𝜈 = 0 we obtain

𝑆NMG =

∫d3𝑥√−𝑔

[−𝑀3𝑅−

√𝑀3𝑞

𝜇𝜈𝐺𝜇𝜈 (13.10)

− 1

4

((𝑚𝑞𝜇𝜈 +∇𝜇𝐴𝜈 +∇𝜈𝐴𝜇 +

2

𝑚∇𝜇∇𝜈𝜋)2

− (𝑚𝑞 + 2∇𝐴+2

𝑚2𝜋)2

)].

Although this action contains apparently higher order terms due to its dependence on ∇𝜇∇𝜈𝜋,this dependence is Galileon-like in that the equations of motion for all fields are second order. Forinstance the naively dangerous combination

(∇𝜇∇𝜈𝜋)2 − (2𝜋)2 (13.11)

is up to a boundary term equivalent to 𝑅𝜇𝜈∇𝜇∇𝜈𝜋. In [141] it is shown that the resulting equationsof motion of all fields are second order due to these special Fierz–Pauli combinations.

As a result of the introduction of the new gauge symmetries, we straightforwardly count thenumber of non-perturbative degrees of freedom. The total number of fields are 16: six from 𝑔𝜇𝜈 ,six from 𝑞𝜇𝜈 , three from 𝐴𝜇 and one from 𝜋. The total number of gauge symmetries are 7:three from diffeomorphisms, three from linear diffeomorphisms and one from the 𝑈(1). Thus, the



Massive Gravity 143

total number of degrees of freedom are 16 − 7 (gauge) − 7 (constraint) = 2 which agrees with thelinearized analysis. An independent argument leading to the same result is given in [317] whereNMG including its topologically massive extension (see below) are presented in Hamiltonian formusing Einstein–Cartan language (see also [176]).

13.3 Decoupling limit of new massive gravity

The formalism of Section 13.2 is also useful for deriving the decoupling limit of NMG which asin the higher dimensional case, determines the leading interactions for the helicity-0 mode. Thedecoupling limit [141] is defined as the limit

𝑀3 →∞ , 𝑚→ 0 Λ5/2 = (√𝑀3𝑚

2)2/5 = fixed . (13.12)

As usual the metric is scaled as

𝑔𝜇𝜈 = 𝜂𝜇𝜈 +1√𝑀3

ℎ𝜇𝜈 , (13.13)

and in the action

𝑆NMG =

∫d3𝑥√−𝑔

[−𝑀3𝑅−

√𝑀3𝑞

𝜇𝜈𝐺𝜇𝜈 (13.14)

− 1

4

((𝑚𝑞𝜇𝜈 +∇𝜇𝐴𝜈 +∇𝜈𝐴𝜇 +

2

𝑚∇𝜇∇𝜈𝜋)2

− (𝑚𝑞 + 2∇𝐴+2

𝑚2𝜋)2

)],

the normalizations have been chosen so that we keep 𝐴𝜇 and 𝜋 fixed in the limit. We readily find

𝑆dec =

∫d3𝑥

[+

1

2ℎ𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 − 𝑞𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 − 𝑞𝜇𝜈(𝜕𝜇𝜕𝜈𝜋 − 𝜂𝜇𝜈2𝜋)

−1

4𝐹𝜇𝜈𝐹

𝜇𝜈 +1

Λ5/25/2

ℰ𝜇𝜈𝛼𝛽ℎ𝛼𝛽(𝜕𝜇𝜋𝜕𝜈𝜋 − 𝜂𝜇𝜈(𝜕𝜋)2)

], (13.15)

where all raising and lowering is understood with respect to the 3 dimensional Minkowski metric.Performing the field redefinition ℎ𝜇𝜈 = 2𝜋𝜂𝜇𝜈 + ℎ𝜇𝜈 + 𝑞𝜇𝜈 we finally obtain

𝑆dec =

∫d3𝑥

[+

1


1

2𝑞𝜇𝜈 ℰ𝛼𝛽𝜇𝜈 𝑞𝛼𝛽

−1

4𝐹𝜇𝜈𝐹

𝜇𝜈 − 1

2(𝜕𝜋)2 − 1

2Λ5/25/2

(𝜕𝜋)22𝜋

]. (13.16)

Thus, we see that in the decoupling limit, NMG becomes equivalent to two massless gravitonswhich have no degrees of freedom, one massless spin-1 particle which has one degree of freedom,and one scalar 𝜋 which has a cubic Galileon interaction. This confirms that the strong couplingscale for NMG is Λ5/2.

The decoupling limit clarifies one crucial aspect of NMG. It has been suggested that NMGcould be power counting renormalizable following previous arguments for topological massivegravity [196] due to the softer nature of divergences in three-dimensional and the existence ofa dimensionless combination of the Planck mass and the graviton mass. This is in fact clearly



144 Claudia de Rham

not the case since the above cubic interaction is a non-renormalizable operator and dominates theFeynman diagrams leading to perturbative unitarity violation at the strong coupling scale Λ5/2

(see Section 10.5 for further discussion on the distinction between the breakdown of perturbativeunitarity and the breakdown of the theory).

13.4 Connection with bi-gravity

The existence of the NMG theory at first sight appears to be something of an anomaly thatcannot be reproduced in higher dimensions. There also does not at first sight seem to be anyobvious connection with the diffeomorphism breaking ghost-free massive gravity model (or dRGT)and multi-gravity extensions. However, in [425] it was shown that NMG, and certain extensionsto it, could all be obtained as scaling limits of the same 3-dimensional bi-gravity models that areconsistent with ghost-free massive gravity in a different decoupling limit. As we already mentioned,the key to seeing this is the auxiliary formulation where the tensor 𝑓𝜇𝜈 is related to the missingextra metric of the bi-gravity theory.

Starting with the 3-dimensional version of bi-gravity [293] in the form

𝑆 =

∫d3𝑥

[𝑀𝑔

2

√−𝑔𝑅[𝑔] + 𝑀𝑓

2

√−𝑓𝑅[𝑓 ]−𝑚2𝒰 [𝑔, 𝑓 ]

], (13.17)

where the bi-gravity potential takes the standard form in terms of characteristic polynomialssimilarly as in (6.4)

𝒰 [𝑔, 𝑓 ] = −𝑀eff

4

3∑𝑛=0

𝛼𝑛ℒ𝑛(𝒦) , (13.18)

and 𝒦 is given in (6.7) in terms of the two dynamical metrics 𝑔 and 𝑓 . The scale 𝑀eff is definedas 𝑀−1

eff =𝑀−1𝑔 +𝑀−1

𝑓 . The idea is to define a scaling limit [425] as follows

𝑀𝑓 → +∞ (13.19)

keeping 𝑀3 = −(𝑀𝑔 +𝑀𝑓 ) fixed and keeping 𝑞𝜇𝜈 fixed in the definition

𝑓𝜇𝜈 = 𝑔𝜇𝜈 −𝑀3

𝑀𝑓𝑞𝜇𝜈 . (13.20)

Since 𝒦𝜇𝜈 → 𝑀3

2𝑀𝑓𝑞𝜇𝜈 , then we have in the limit

𝑆 =

∫d3𝑥

[− 𝑀3

2

√−𝑔𝑅[𝑔]− 𝑀3

2𝑞𝜇𝜈𝐺𝜇𝜈(𝑔) +

𝑚2𝑀eff

4

3∑𝑛=0

𝛼𝑛(−1)𝑛(𝑀3

𝑀𝑓

)𝑛ℒ𝑛(𝑞)

]which prompts the definition of a new set of coefficients

𝑐𝑛 = − (−1)𝑛

2𝑀3��𝛼𝑛

(𝑀3

𝑀𝑓

)𝑛, (13.21)

so that

𝑆 =𝑀3

∫d3𝑥

[−√−𝑔𝑅[𝑔]− 𝑞𝜇𝜈𝐺𝜇𝜈(𝑔)−𝑚2

3∑𝑛=0

𝑐𝑛ℒ𝑛(𝑞)

]. (13.22)

Since this theory is obtained as a scaling limit of the ghost-free bi-gravity action, it is guaranteedto be free from the BD ghost. We see that in the case 𝑐2 = 1/4, 𝑐3 = 𝑐4 = 0 we obtain theauxiliary field formulation of NMG, justifying the connection between the auxiliary field 𝑞𝜇𝜈 andthe bi-gravity metric 𝑓𝜇𝜈 .



Massive Gravity 145

13.5 3D massive gravity extensions

The generic form of the auxiliary field formulation of NMG derived above [425]

𝑆 =𝑀3

∫d3𝑥

[−√−𝑔𝑅[𝑔]− 𝑞𝜇𝜈𝐺𝜇𝜈(𝑔)−𝑚2

3∑𝑛=0

𝑐𝑛ℒ𝑛(𝑞)

], (13.23)

demonstrates that there exists a two parameter family extensions of NMG determined by nonzerocoefficients for 𝑐3 and 𝑐4. The purely metric formulation for the generic case can be determined byintegrating out the auxiliary field 𝑞𝜇𝜈 . The equation of motion for 𝑞𝜇𝜈 is given symbolically

−𝐺−𝑚23∑

𝑛=1

𝑛𝑐𝑛 𝜖 𝜖 𝑞𝑛−1𝑔3−𝑛 = 0 . (13.24)

This is a quadratic equation for the tensor 𝑞𝜇𝜈 . Together, these two additional degrees of free-dom give the cubic curvature [447] and Born–Infeld extension NMG [279]. Although additionalhigher derivative corrections have been proposed based on consistency with the holographic c-theorem [424], the above connection suggests that Eq. (13.23) is the most general set of interactionsallowed in NMG which are free from the BD ghost.

In the specific case of the Born–Infeld extension [279] the action is

𝑆B.I = 4𝑚2𝑀3

∫d3𝑥

[√−𝑔 −

√−det[𝑔𝜇𝜈 −

1

𝑚2𝐺𝜇𝜈 ]

]. (13.25)

It is straightforward to show that on expanding the square root to second order in 1/𝑚2 werecover the original NMG action. The specific case of the Born–Infeld extension of NMG, alsohas a surprising role as a counterterm in the AdS4 holographic renormalization group [329]. Thesignificance of this relation is unclear at present.

13.6 Other 3D theories

13.6.1 Topological massive gravity

In four dimensions, the massive spin-2 representations of the Poincare group must come in positiveand negative helicity pairs. By contrast, in three dimensions the positive and negative helicitystates are completely independent. Thus, while a parity preserving theory of massive gravity inthree dimensions will contain two propagating degrees of freedom, it seems possible in principle forthere to exist an interacting theory for one of the helicity modes alone. What is certainly possibleis that one can give different interactions to the two helicity modes. Such a theory necessarilybreaks parity, and was found in [180, 179]. This theory is known as ‘topologically massive gravity’(TMG) and is described by the Einstein–Hilbert action, with cosmological constant, supplementedby a term constructed entirely out of the connection (hence the name topological)

𝑆 =𝑀3

2

∫d3𝑥√−𝑔(𝑅− 2Λ) +

1

4𝜇𝜖𝜆𝜇𝜈Γ𝜌𝜆𝜎

[𝜕𝜇Γ

𝜎𝜌𝜈 +

2

3Γ𝜎𝜇𝜏Γ

𝜏𝜈𝜌

]. (13.26)

The new interaction is a gravitational Chern–Simons term and is responsible for the parity breaking.More generally, this action may be supplemented to the NMG Lagrangian interactions and so theTMG can be viewed as a special case of the full extended parity violating NMG.

The equations of motion for topologically massive gravity take the form

𝐺𝜇𝜈 + Λ𝑔𝜇𝜈 +1

𝜇𝐶𝜇𝜈 = 0 , (13.27)



146 Claudia de Rham

where 𝐶𝜇𝜈 is the Cotton tensor which is given by

𝐶𝜇𝜈 = 𝜖𝜇𝛼𝛽∇𝛼(𝑅𝛽𝜈 −

1

4𝑔𝛽𝜈𝑅) . (13.28)

Einstein metrics for which 𝐺𝜇𝜈 = −Λ𝑔𝜇𝜈 remain as a subspace of general set of vacuum solutions.In the case where the cosmological constant is negative Λ = −1/ℓ2 we can use the correspondenceof Brown and Henneaux [78] to map the theory of gravity on an asymptotically AdS3 space to a2D CFT living at the boundary.

The AdS/CFT in the context of topological massive gravity was also studied in Ref. [449].

13.6.2 Supergravity extensions

As with any gravitational theory, it is natural to ask whether extensions exist which exhibit localsupersymmetry, i.e., supergravity. A supersymmetric extension to topologically massive gravitywas given in [182]. An 𝑁 = 1 supergravity extension of NMG including the topologically massivegravity terms was given in [21] and further generalized in [67]. The construction requires theintroduction of an ‘auxiliary’ bosonic scalar field 𝑆 so that the form of the action is

𝑆 =1

𝜅2

∫d3𝑥√−𝑔[𝑀ℒ𝐶 + 𝜎ℒ𝐸.𝐻. +

1

𝑚2ℒ𝐾 +

1

8��2ℒ𝑅2 +

1

��2ℒ𝑆4 +

1

��ℒ𝑆3

]+

∫d3𝑥

1

𝜇ℒtop , (13.29)

where

ℒ𝐶 = 𝑆 + fermions (13.30)

ℒE.H. = 𝑅− 2𝑆2 + fermions (13.31)

ℒ𝐾 = 𝐾 − 1

2𝑆2𝑅− 3

2𝑆4 + fermions (13.32)

ℒ𝑅2 = −16[(𝜕𝑆)2 − 9

4(𝑆2 +

1

6𝑅)2

]+ fermions (13.33)

ℒ𝑆4 = 𝑆4 +3

10𝑅𝑆2 + fermions (13.34)

ℒ𝑆3 = 𝑆3 +1

2𝑅𝑆 + fermions (13.35)

ℒtop =1

4𝜖𝜆𝜇𝜈Γ𝜌𝜆𝜎

[𝜕𝜇Γ

𝜎𝜌𝜈 +

2

3Γ𝜎𝜇𝜏Γ

𝜏𝜈𝜌

]+ fermions . (13.36)

The fermion terms complete each term in the Lagrangian into an independent supersymmetricinvariant. In other words supersymmetry alone places no further restrictions on the parametersin the theory. It can be shown that the theory admits supersymmetric AdS vacua [21, 67]. Theextensions of this supergravity theory to larger numbers of supersymmetries is considered at thelinearized level in [64].

Moreover, 𝑁 = 2 supergravity extensions of TMG were recently constructed in Ref. [370] andits 𝑁 = 3 and 𝑁 = 4 supergravity extensions in Ref. [371].

13.6.3 Critical gravity

Finally, let us comment on a special case of three dimensional gravity known as log gravity [65] orcritical gravity in analogy with the general dimension case [384, 183, 16]. For a special choice of



Massive Gravity 147

parameters of the theory, there is a degeneracy in the equations of motion for the two degrees offreedom leading to the fact that one of the modes of the theory becomes a ‘logarithmic’ mode.

Indeed, at the special point 𝜇ℓ = 1, (where ℓ is the AdS length scale, Λ = −1/ℓ2), known asthe ‘chiral point’ the left-moving (in the language of the boundary CFT) excitations of the theorybecome pure gauge and it has been argued that the theory then becomes purely an interactingtheory for the right moving graviton [93]. In Ref. [377] it was earlier argued that there was nomassive graviton excitations at the critical point 𝜇ℓ = 1, however Ref. [93] found one massivegraviton excitation for every finite and non-zero value of 𝜇ℓ, including at the critical point 𝜇ℓ = 1.

This case was further analyzed in [273], see also Ref. [274] for a recent review. It was shownthat the degeneration of the massive graviton mode with the left moving boundary graviton leadsto logarithmic excitations.

To be more precise, starting with the auxiliary formulation of NMGwith a cosmological constant𝜆𝑚2

𝑆NMG =𝑀3

∫d3𝑥√−𝑔[𝜎𝑅− 2𝜆𝑚2 − 𝑞𝜇𝜈𝐺𝜇𝜈 −

1


𝜇𝜈 − 𝑞2)], (13.37)

we can look for AdS vacuum solutions for which the associated cosmological constant Λ = −1/ℓ2in 𝐺𝜇𝜈 = −Λ𝑔𝜇𝜈 is not the same as 𝜆𝑚2. The relation between the two is set by the vacuumequations to be

− 1

4𝑚2Λ2 − Λ𝜎 + 𝜆𝑚2 = 0 , (13.38)

which generically has two solutions. Perturbing the action to quadratic order around this vacuumsolution we have

𝑆2 =𝑀3

∫d3𝑥

[− ��2ℎ𝜇𝜈𝒢𝜇𝜈 − 𝑞𝜇𝜈𝒢𝜇𝜈 −

1


𝜇𝜈 − 𝑞2)]. (13.39)

where𝒢𝜇𝜈(ℎ) = ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 − 2Λℎ𝜇𝜈 + Λ𝑔𝜇𝜈ℎ (13.40)

and

�� = 𝜎 − Λ

3𝑚2(13.41)

where we raise and lower the indices with respect to the background AdS metric 𝑔𝜇𝜈 .As usual, it is apparent that this theory describes one massless graviton (with no propagating

degrees of freedom) and one massive one whose mass is given by 𝑀2 = −𝑚2��. However, bychoosing �� = 0 the massive mode becomes degenerate with the existing massless one.

In this case, the action is

𝑆2 =𝑀3

∫d3𝑥

[−𝑞𝜇𝜈𝒢𝜇𝜈 −

1


𝜇𝜈 − 𝑞2)], (13.42)

and varying with respect to ℎ𝜇𝜈 and 𝑞𝜇𝜈 we obtain the equations of motion

𝒢𝜇𝜈(𝑞) = 0 , (13.43)

𝒢𝜇𝜈(ℎ) +1

2(𝑞𝜇𝜈 − 𝑔𝜇𝜈𝑞) = 0 . (13.44)

Choosing the gauge ∇𝜇ℎ𝜇𝜈 − ∇𝜈ℎ = 0, the equations of motion imply ℎ = 0 and the resultingequation of motion for ℎ𝜇𝜈 takes the form

[2− 2Λ]2ℎ𝜇𝜈 = 0 . (13.45)



148 Claudia de Rham

It is this factorization of the equations of motion into a square of an operator that is characteristic ofthe critical/log gravity theories. Although the equation of motion is solved by the usual masslessmodels for which [2− 2Λ]ℎ𝜇𝜈 = 0, there are additional logarithmic modes which do not solvethis equation but do solve Eq. (13.45). These are so-called because they behave logarithmicallyin 𝜌 asymptotically when the AdS metric is put in the form d𝜌2 = ℓ2(− cosh(𝜌)2 d𝜏2 + d𝜌2 +sinh(𝜌)2 d𝜃2). The presence of these log modes was shown to remain beyond the linear regime, seeRef. [271].

Based on this result as well as on the finiteness and conservation of the stress tensor and on theemergence of a Jordan cell structure in the Hamiltonian, the correspondence to a logarithmic CFTwas conjectured in Ref. [273], where the to be dual log CFTs representations have degeneracies inthe spectrum of scaling dimensions.

Strong indications for this correspondence appeared in many different ways. First, consistentboundary conditions which allow the log modes were provided in Ref. [272], were it was shown thatin addition to the Brown–Henneaux boundary conditions one could also consider more general ones.These boundary conditions were further explored in [305, 397], where it was shown that the stress-energy tensor for these boundary conditions are finite and not chiral, giving another indicationthat the theory could be dual to a logarithmic CFT.

Then specific correlator functions were computed and compared. Ref. [449] checked the 2-pointcorrelators and Ref. [275] the 3-point ones. A similar analysis was also performed within thecontext of NMG in Ref. [270] where the 2-point correlators were computed at the chiral point andshown to behave as those of a logarithmic CFT.

Further checks for this AdS/log CFT include the 1-loop partition function as computed inRef. [241]. See also Ref. [274] for a review of other checks.

It has been shown, however, that ultimately these theories are non-unitary due to the fact thatthere is a non-zero inner product between the log modes and the normal models and the inabilityto construct a positive definite norm on the Hilbert space [432].

13.7 Black holes and other exact solutions

A great deal of physics can be learned from studying exact solutions, in particular those corre-sponding to black hole geometries. Black holes are also important probes of the non-perturbativeaspects of gravitational theories. We briefly review here the types of exact solutions obtained inthe literature.

In the case of topologically massive gravity, a one-parameter family of extensions to the BTZblack hole have been obtained in [245]. In the case of NMG as well as the usual BTZ black holesobtained in the presence of a negative cosmological constant there are in a addition a class ofwarped AdS3 black holes [102] whose metric takes the form

d𝑠2 = −𝛽2 𝜌2 − 𝜌20𝑟2

d𝑡2 + 𝑟2(d𝜑− 𝜌+ (1− 𝛽2)𝜔

𝑟2d𝑡

)2

+1

𝛽2𝜁2d𝜌2

𝜌2 − 𝜌20, (13.46)

where the radial coordinate 𝑟 is given by

𝑟2 = 𝜌2 + 2𝜔𝜌+ 𝜔2(1− 𝛽2) +𝛽2𝜌201− 𝛽2

. (13.47)

and the parameters 𝛽 and 𝜁 are determined in terms of the graviton mass 𝑚 and the cosmologicalconstant Λ by

𝛽2 =9− 21Λ/𝑚2 ∓ 2

√3(5 + 7Λ/𝑚2

4(1− Λ/𝑚2), 𝜁−2 =

21− 4𝛽2

8𝑚2. (13.48)



Massive Gravity 149

This metric exhibits two horizons at 𝜌 = ±𝜌0, if 𝛽2 ≥ 0 and 𝜌0 is real. Absence of closed timelikecurves requires that 𝛽2 ≤ 1. This puts the allowed range on the values of Λ to be

−35𝑚2

289≤ Λ ≤ 𝑚2

21. (13.49)

AdS waves, extensions of plane (pp) waves anti-de Sitter spacetime have been considered in [33].Further work on extensions to black hole solutions, including charged black hole solutions can befound in [418, 101, 253, 5, 6, 372, 427, 250]. We note in particular the existence of a class of Lifshitzblack holes [32] that exhibit the Lifshitz anisotropic scale symmetry

𝑡→ 𝜆𝑧𝑡 , ��→ 𝜆�� , (13.50)

where 𝑧 is the dynamical critical exponent. As an example for 𝑧 = 3 the following Lifshitz blackhole can be found [32]

d𝑠2 = −𝑟6

ℓ6

(1− 𝑀𝑙2

𝑟2

)d𝑡2 +

d𝑟2(𝑟2

ℓ2 −𝑀) + 𝑟2 d𝜑2 . (13.51)

This metric has a curvature singularity at 𝑟 = 0 and a horizon at 𝑟+ = ℓ√𝑀 . The Lifshitz

symmetry is preserved if we scale 𝑡 → 𝜆3𝑡, 𝑥 → 𝜆𝑥, 𝑟 → 𝜆−1𝑟 and in addition we scale the blackhole mass as𝑀 → 𝜆−2𝑀 . The metric should be contrasted with the normal BTZ black hole whichcorresponds to 𝑧 = 1

d𝑠2 = −𝑟2

ℓ2

(1− 𝑀ℓ2

𝑟2

)d𝑡2 +

d𝑟2(𝑟2

ℓ2 −𝑀) + 𝑟2 d𝜑2 . (13.52)

Exact solutions for charged black holes were also derived in Ref. [249] and an exact, non-stationarysolution of TMG and NMG with the asymptotic charges of a BTZ black hole was find in [227].This exact solution was shown to admit a timelike singularity. Other exact asymptotically AdS-likesolutions were found in Ref. [251].

13.8 New massive gravity holography

One of the most interesting avenues of exploration for NMG has been in the context of Maldacena’sAdS/CFT correspondence [396]. According to this correspondence, NMG with a cosmologicalconstant chosen so that there are asymptotically anti-de Sitter solutions is dual to a conformal fieldtheory (CFT). This has been considered in [67, 381, 380] where it was found that the requirementsof bulk unitarity actually lead to a negative central charge.

The argument for this proceeds from the identification of the central charge of the dual twodimensional field theory with the entropy of a black hole in the bulk using Cardy’s formula. Theentropy of the black hole is given by [368]

𝑆 =𝐴BTZ

4𝐺3Ω , (13.53)

where 𝐺3 is the 3-dimensional Newton constant and Ω = 2𝐺3

3ℓ 𝑐 where 𝑙 is the AdS radius and 𝑐is the central charge. This formula is such that 𝑐 = 1 for pure Einstein–Hilbert gravity with anegative cosmological constant.

A universal formula for this central charge has been obtained as is given by

𝑐 =ℓ

2𝐺3𝑔𝜇𝜈

𝜕ℒ𝜕𝑅𝜇𝜈

. (13.54)



150 Claudia de Rham

This result essentially follows from using the Wald entropy formula [480] for a higher derivativegravity theory and identifying this with the central change through the Cardy formula. Applyingthis argument for new massive gravity we obtain [67]

𝑐 =3ℓ

2𝐺3

(𝜎 +

1

2𝑚2ℓ2

). (13.55)

Since 𝜎 = −1 is required for bulk unitarity, we must choose 𝑚2 > 0 to have a chance of getting 𝑐positive. Then we are led to conclude that the central charge is only positive if

Λ = − 1

ℓ2< −2𝑚2 . (13.56)

However, unitarity in the bulk requires 𝑚2 > −Λ/2 and this excludes this possibility. We are thusled to conclude that NMG cannot be unitary both in the bulk and in the dual CFT. This failureto maintain both bulk and boundary unitarity can be resolved by a modification of NMG to a fullbi-gravity model, namely Zwei-Dreibein gravity to which we turn next.

13.9 Zwei-dreibein gravity

As we have seen, there is a conflict in NMG between unitarity in the bulk, i.e., the requirement thatthe massive gravitons are not ghosts, and unitarity in dual CFT as required by the positivity of thecentral charge. This conflict may be resolved, however, by replacing NMG with the 3-dimensionalbi-gravity extension of ghost-free massive gravity that we have already discussed. In particular,if we work in the Einstein–Cartan formulation in three dimensions, then the metric is replacedby a ‘dreibein’ and since this is a bi-gravity model, we need two ‘dreibeins’. This gives us theZwei-dreibein gravity [63].

In the notation of [63] the Lagrangian is given by

ℒ = −𝜎𝑀1𝑒𝑎𝑅𝑎(𝑒)−𝑀2𝑓𝑎𝑅

𝑎(𝑓)− 1

6𝑚2𝑀1𝛼1𝜖𝑎𝑏𝑐𝑒

𝑎𝑒𝑏𝑒𝑐

−1

6𝑚2𝑀2𝛼2𝜖𝑎𝑏𝑐𝑓

𝑎𝑓 𝑏𝑓 𝑐 +1

2𝑚2𝑀12𝜖𝑎𝑏𝑐(𝛽1𝑒

𝑎𝑒𝑏𝑓 𝑐 + 𝛽2𝑒𝑎𝑓 𝑏𝑓 𝑐) (13.57)

where we have suppressed the wedge products 𝑒3 = 𝑒∧𝑒∧𝑒, 𝑅𝑎(𝑒) is Lorentz vector valued curvaturetwo-form for the spin-connection associated with the dreibein 𝑒 and 𝑅𝑎(𝑓) that associated withthe dreibein 𝑓 . Since we are in three dimensions, the spin-connection can be written as a Lorentzvector dualizing with the Levi-Civita symbol 𝜔𝑎 = 𝜖𝑎𝑏𝑐𝜔𝑏𝑐. This is nothing other than the vierbeinrepresentation of bi-gravity with the usual ghost-free (dRGT) mass terms. As we have alreadydiscussed, NMG and its various extensions arise in appropriate scaling limits.

A computation of the central charge following the same procedure was given in [63] with theresult that

𝑐 = 12𝜋ℓ(𝜎𝑀1 + 𝛾𝑀2) . (13.58)

Defining the parameter 𝛾 via the relation

(𝛼2(𝜎𝑀1 +𝑀2) + 𝛽2𝑀2)𝛾2 + 2(𝑀2𝛽1 − 𝜎𝑀1𝛽2)𝛾

−𝜎(𝛼1(𝜎𝑀1 +𝑀2) + 𝛽1𝑀1) = 0 , (13.59)

then bulk unitarity requires 𝛾/(𝜎𝑀1+𝛾𝑀2) < 0. In order to have 𝑐 > 0 we thus need 𝛾 < 0 whichin turn implies 𝜎 > 1 (since 𝑀1 and 𝑀2 are defined as positive). The absence of tachyons in theAdS vacuum requires 𝛽1 + 𝛾𝛽2 > 0, and this assumes a real solution for 𝛾 for a negative Λ. Thereare an open set of such solutions to these conditions, which shows that the conditions for unitarity



Massive Gravity 151

are not finely tuned. For example in [63] it is shown that there is an open set of solutions whichare close to the special case 𝑀1 = 𝑀2, 𝛽1 = 𝛽2 = 1, 𝛾 = 1 and 𝛼1 = 𝛼2 = 3/2 + 1

ℓ2𝑚2 . Thisresult is not in contradiction with the scaling limit that reproduces NMG, because this scalinglimit requires the choice 𝜎 = −1 which is in contradiction with positive central charge.

These results potentially have an impact on the higher dimensional case. We see that inthree dimensions we potentially have a diffeomorphism invariant theory of massive gravity (i.e.,bi-gravity) which at least for AdS solutions exhibits unitarity both in the bulk and in the boundaryCFT for a finite range of parameters in the theory. However, these bi-gravity models are easilyextended into all dimensions as we have already discussed and it is similarly easy to find AdSsolutions which exhibit bulk unitarity. It would be extremely interesting to see if the associated dualCFTs are also unitary thus providing a potential holographic description of generalized theories ofmassive gravity.

14 Lorentz-Violating Massive Gravity

14.1 SO(3)-invariant mass terms

The entire analysis performed so far is based on assuming Lorentz invariance. In what follows webriefly review a few other potentially viable theories of massive gravity where Lorentz invarianceis broken and their respective cosmology.

Prior to the formulation of the ghost-free theory of massive gravity, it was believed that noLorentz invariant theories of massive gravity could evade the BD ghost and Lorentz-violatingtheories were thus the best hope; we refer the reader to [438] for a thorough review on the field.A thorough analysis of Lorentz-violating theories of massive gravity was performed in [200] andmore recently in [107]. See also Refs. [236, 73, 114] for other complementary studies. Since thisfield has been reviewed in [438] we only summarize the key results in this section (see also [74] fora more recent review on many developments in Lorentz violating theories.) See also Ref. [378] foran interesting spontaneous breaking of Lorentz invariance in ghost-free massive gravity using threescalar fields, and Ref. [379] for a 𝑆𝑂(3)-invariant ghost-free theory of massive gravity which canbe formulated with three Stuckelberg scalar fields and propagating five degrees of freedom.

In most theories of Lorentz-violating massive gravity, the 𝑆𝑂(3, 1) Poincare group is brokendown to a 𝑆𝑂(3) rotation group. This implies the presence of a preferred time. Preferred-frameeffects are, however, strongly constrained by solar system tests [487] as well as pulsar tests [54],see also [494, 493] for more recent and even tighter constraints.

At the linearized level the general mass term that satisfies this rotation symmetry is

ℒ𝑆𝑂(3) mass =1

8

(𝑚2

0ℎ200 + 2𝑚2

1ℎ20𝑖 −𝑚2

2ℎ2𝑖𝑗 +𝑚2

3ℎ𝑖𝑖ℎ𝑗𝑗 − 2𝑚2

4ℎ00ℎ𝑖𝑖

), (14.1)

where space indices are raised and lowered with respect to the flat spatial metric 𝛿𝑖𝑗 . This extendsthe Lorentz invariant mass term presented in (2.39). In the rest of this section, we will establishthe analogue of the Fierz–Pauli mass term (2.44) in this Lorentz violating case and establish theconditions on the different mass parameters 𝑚0,1,2,3,4.

We note that Lorentz invariance is restored when 𝑚1 = 𝑚2, 𝑚3 = 𝑚4 and 𝑚20 = −𝑚2

1 +𝑚23.

The Fierz–Pauli structure then further fixes 𝑚1 = 𝑚3 implying 𝑚0 = 0, which is precisely whatensures the presence of a constraint and the absence of BD ghost (at least at the linearized level).

Out of these five mass parameters some of them have a direct physical meaning [200, 437, 438]

∙ The parameter 𝑚2 is the one that represents the mass of the helicity-2 mode. As a result weshould impose 𝑚2

2 ≥ 0 to avoid tachyon-like instabilities. Although we should bear in mindthat if that mass parameter is of the order of the Hubble parameter today 𝑚2 ≃ 10−33 eV,then such an instability would not be problematic.



152 Claudia de Rham

∙ The parameter 𝑚1 is the one responsible for turning on a kinetic term for the two helicity-1modes. Since 𝑚1 = 𝑚2 in a Lorentz-invariant theory of massive gravity, the helicity-1 modecannot be turned off (𝑚1 = 0) while maintaining the graviton massive (𝑚2 = 0). This is astandard result of Lorentz invariant massive gravity seen so far where the helicity-1 mode isalways present. For Lorentz breaking theories the theory is quite different and one can easilyswitch off at the linearized level the helicity-1 modes in a theory of Lorentz-breaking massivegravity. The absence of a ghost in the helicity-1 mode requires 𝑚2

1 ≥ 0.

∙ If 𝑚0 = 0 and 𝑚1 = 0 and 𝑚4 = 0 then two scalar degrees of freedom are present alreadyat the linear level about flat space-time and one of these is always a ghost. The absence ofghost requires either 𝑚0 = 0 or 𝑚1 = 0 or finally 𝑚4 = 0 and 𝑚2 = 𝑚3.

In the last scenario, where 𝑚4 = 0 and 𝑚2 = 𝑚3, the scalar degree of freedom loses itsgradient terms at the linear level, which means that this mode is infinitely strongly coupledunless no gradient appears fully non-linearly either.

The case 𝑚0 has an interesting phenomenology as will be described below. While it prop-agates five degrees of freedom about Minkowski it avoids the vDVZ discontinuity in aninteresting way.

Finally, the case 𝑚1 = 0 (including when 𝑚0 = 0) will be discussed in more detail in whatfollows. It is free of both scalar (and vector) degrees of freedom at the linear level aboutMinkowski and thus evades the vDVZ discontinuity in a straightforward way.

∙ The analogue of the Higuchi bound was investigated in [72]. In de Sitter with constantcurvature 𝐻, the generalized Higuchi bound is

𝑚44 + 2𝐻2

(3(𝑚2

3 −𝑚24)−𝑚2

2

)> 𝑚2

4(𝑚21 −𝑚2

4) if 𝑚0 = 0 , (14.2)

while if instead 𝑚1 = 0 then no scalar degree of freedom are propagating on de Sitter eitherso there is no analogue of the Higuchi bound (a scalar starts propagating on FLRW solutionsbut it does not lead to an equivalent Higuchi bound either. However, the absence of tachyonand gradient instabilities do impose some conditions between the different mass parameters).

As shown in the case of the Fierz–Pauli mass term and its non-linear extension, one of the mostnatural way to follow the physical degrees of freedom and their health is to restore the brokensymmetry with the appropriate number of Stuckelberg fields.

In Section 2.4 we reviewed how to restore the broken diffeomorphism invariance using fourStuckelberg fields 𝜑𝑎 using the relation (2.75). When Lorentz invariance is broken, the Stuckelbergtrick has to be performed slightly differently. Performing an ADM decomposition, which is ap-propriate for the type of Lorentz breaking we are considering, we can use for Stuckelberg scalarfields Φ = Φ0 and Φ𝑖, 𝑖 = 1, · · · , 3 to define the following four-dimensional scalar, vector andtensors [107]

𝑛 = (−𝑔𝜇𝜈𝜕𝜇Φ𝜕𝜈Φ)−1/2 (14.3)

𝑛𝜇 = 𝑛𝜕𝜇Φ (14.4)

𝑌 𝜇𝜈 = 𝑔𝜇𝛼𝜕𝛼Φ𝑖𝜕𝜈Φ

𝑗𝛿𝑖𝑗 (14.5)

Γ𝜇𝜈 = 𝑌 𝜇𝜈 + 𝑛𝜇𝑛𝛼𝜕𝛼Φ𝑖𝜕𝜈Φ

𝑗𝛿𝑖𝑗 . (14.6)

𝑛 can be thought of as the ‘Stuckelbergized’ version of the lapse and Γ𝜇𝜈 as that of the spatialmetric.

In the Lorentz-invariant case we are stuck with the combination 𝑋𝜇𝜈 = 𝑌 𝜇𝜈 −𝑛−2𝑔𝜇𝛼𝑛𝛼𝑛𝜈 , but

this combination can be broken here and the mass term can depend separately on 𝑛, 𝑛𝜇 and 𝑌 .



Massive Gravity 153

This allows for new mass terms. In [107] this framework was derived and used to find new massterms that exhibit five degrees of freedom. This formalism was also developed in [200] and usedto derive new mass terms that also have fewer degrees of freedom. We review both cases in whatfollows.

14.2 Phase 𝑚1 = 0

14.2.1 Degrees of freedom on Minkowski

As already mentioned, the helicity-1 mode have no kinetic term at the linear level on Minkowskiif 𝑚1 = 0. Furthermore, it turns out that the field 𝑣 in (14.17) is the Lagrange multiplier whichremoves the BD ghost (as opposed to the field 𝜓 in the case 𝑚0 = 0 presented previously). Itimposes the constraint 𝜏 = 0 which in turns implies 𝜏 = 0. Using this constraint back in the action,one can check that there remains no time derivatives on any of the scalar fields, which means thatthere are no propagating helicity-0 mode on Minkowski either [200, 437, 438]. So in the case where𝑚1 = 0 there are only 2 modes propagating in the graviton on Minkowski, the 2 helicity-2 modesas in GR.

In this case, the absence of the ghost can be seen to follow from the presence of a residualsymmetry on flat space [200, 438]

𝑥𝑖 → 𝑥𝑖 + 𝜉𝑖(𝑡) , (14.7)

for three arbitrary functions 𝜉𝑖(𝑡). In the Stuckelberg language this implies the following internalsymmetry

Φ𝑖 → Φ𝑖 + 𝜉𝑖(Φ) . (14.8)

To maintain this symmetry non-linearly the mass term should be a function of 𝑛 and Γ𝜇𝜈 [438]

ℒmass = −𝑚2𝑀2

Pl

8

√−𝑔 𝐹 (𝑛,Γ𝜇𝜈) . (14.9)

The absence of helicity-1 and -0 modes while keeping the helicity-2 mode massive makes thisLorentz violating theory of gravity especially attractive. Its cosmology was explored in [201] andit turns out that this theory of massive gravity could be a candidate for cold dark matter as shownin [202].

Moreover, explicit black hole solutions were presented in [438] where it was shown that inthis theory of massive gravity black holes have hair and the Stuckelberg fields (in the Stuckelbergformulation of the theory) do affect the solution. This result is tightly linked to the fact that thistheory of massive gravity admits instantaneous interactions which is generic to any action of theform (14.9).

14.2.2 Non-perturbative degrees of freedom

Perturbations on more general FLRW backgrounds were then considered more recently in [72].Unlike in Minkowski, scalar perturbations on curved backgrounds are shown to behave in a similarway for the cases 𝑚1 = 0 and 𝑚0 = 0. However, as we shall see below, the case 𝑚0 = 0 propagatesfive degrees of freedom including a helicity-0 mode that behaves as a scalar it follows that ongeneric backgrounds the theory with 𝑚1 = 0 also propagates a helicity-0 mode. The helicity-0mode is thus infinitely strongly coupled when considered perturbatively about Minkowski.



154 Claudia de Rham

14.3 General massive gravity (𝑚0 = 0)

In [107] the most general mass term that extends (14.1) non-linearly was considered. It can bewritten using the Stuckelberg variables defined in (14.3), (14.4) and (14.5),

ℒSO(3) mass = −𝑚2𝑀2

Pl

8

√−𝑔 𝑉 (𝑛, 𝑛𝜇, 𝑌

𝜇𝜈) . (14.10)

Generalizing the Hamiltonian analysis for this mass term and requiring the propagation of fivedegrees of freedom about any background led to the Lorentz-invariant ghost-free theory of massivegravity presented in Part II as well as two new theories of Lorentz breaking massive gravity.

All of these cases ensures the absence of BD ghost by having 𝑚0 = 0. The case where the BDghost is projected thanks to the requirement 𝑚1 = 0 is discussed in Section 14.2.

14.3.1 First explicit Lorentz-breaking example with five dofs

The first explicit realization of a consistent nonlinear Lorentz breaking model is as follows [107]

𝑉1 (𝑛, 𝑛𝜇, 𝑌𝜇𝜈) = 𝑛−1

[𝑛+ 𝜁(Γ)

]𝑈(��) + 𝑛−1𝒞(Γ) , (14.11)

with

��𝜇𝜈 =

⎛⎜⎝Γ𝜇𝛼 − 𝑛2𝑛𝜇𝑛𝛼[𝑛+ 𝜁(Γ)

]2⎞⎟⎠ 𝜕𝛼Φ

𝑖𝜕𝜈Φ𝑗𝛿𝑖𝑗 , (14.12)

and where 𝑈 , 𝐶 and 𝜁 are scalar functions.The fact that several independent functions enter the mass term will be of great interest for

cosmology as one of these functions (namely 𝒞) can be used to satisfy the Bianchi identity whilethe other function can be used for an appropriate cosmological history.

The special case 𝜁 = 0 is what is referred to as the ‘minimal model’ and was investigatedin [108]. In unitary gauge, this minimal model is simply

ℒ(minimal)SO(3) mass = −

𝑚2𝑀2Pl

8

√−𝑔(𝑈(𝑔𝑖𝑘𝛿𝑘𝑗) +𝑁−1𝒞(𝛾𝑖𝑘𝛿𝑖𝑗)

), (14.13)

where 𝛾𝑖𝑗 is the spatial part of the metric and 𝑔𝑖𝑗 = 𝛾𝑖𝑗 −𝑁−2𝑁 𝑖𝑁 𝑗 , where 𝑁 is the lapse and 𝑁 𝑖

the shift.This minimal model is of special interest as both the primary and secondary second-class

constraints that remove the sixth degree of freedom can be found explicitly and on the constraintsurface the contribution of the mass term to the Hamiltonian is

𝐻 ∝𝑀2Pl𝑚

2

∫d3𝑥√𝛾𝒞(𝛾𝑖𝑘𝛿𝑖𝑗) , (14.14)

where the overall factor is positive so the Hamiltonian is positive definite as long as the function𝒞 is positive.

At the linearized level about Minkowski (which is a vacuum solution) this theory can be param-eterized in terms of the mass scales introduces in (14.1) with 𝑚0 = 0, so the BD ghost is projectedout in a way similar as in ghost-free massive gravity.

Interestingly, if 𝒞 = 0, this theory corresponds to 𝑚1 = 0 (in addition to 𝑚0 = 0) which as seenearlier the helicity-1 mode is absent at the linearized level. However, they survive non-linearly andso the case 𝒞 = 0 is infinitely strongly coupled.



Massive Gravity 155

14.3.2 Second example of Lorentz-breaking with five dofs

Another example of Lorentz breaking 𝑆𝑂(3) invariant theory of massive gravity was providedin [107]. In that case the Stuckelberg language is not particularly illuminating and we simply givethe form of the mass term in unitary gauge (Φ = 𝑡 and Φ𝑖 = 𝑥𝑖),

𝑉2 =𝑐12

[��𝑇 (𝑁 I+M)

−1 (F+𝑁−1MF)(𝑁 I+M)

−1��]

(14.15)

+𝒞 +𝑁−1𝒞 .

where F = {𝑓𝑖𝑗} is the spatial part of the reference metric (for a Minkowski reference metric

𝑓𝑖𝑗 = 𝛿𝑖𝑗), 𝑐1 is a constant, and 𝒞 and 𝒞 are functions of the spatial metric 𝛾𝑖𝑗 , while M is a rank-3matrix which depends on 𝛾𝑖𝑘𝑓𝑘𝑗 .

Interestingly, 𝒞 does not enter the Hamiltonian on the constraint surface. The contribution ofthis mass term to the on-shell Hamitonian is [107]

𝐻 ∝𝑀2Pl𝑚

2

∫d3𝑥√𝛾[−𝑐1

2��𝑇 (𝑁 I+M)

−1 FM (𝑁 I+M)−1�� + 𝒞

], (14.16)

with a positive coefficient, which implies that 𝒞 should be bounded from below

14.3.3 Absence of vDVZ and strong coupling scale

Unlike in the Lorentz-invariant case, the kinetic term for the Stuckelberg fields does not only arisefrom the mixing with the helicity-2 mode.

When looking at perturbations about Minkowski and focusing on the scalar modes we canfollow the analysis of [437],

d𝑠2 = −(1− 𝜓) d𝑡2 + 2𝜕𝑖𝑣 d𝑥𝑖 d𝑡+ (𝛿𝑖𝑗 + 𝜏𝛿𝑖𝑗 + 𝜕𝑖𝜕𝑗𝜎) d𝑥𝑖 d𝑥𝑗 , (14.17)

when 𝑚0 = 0 𝜓 plays the role of the Lagrange multiplier for the primary constraint imposing

𝜎 =

(2

𝑚24

− 3

∇

)𝜏 , (14.18)

where ∇ is the three-dimensional Laplacian. The secondary constraint then imposes the relation

𝑣 =2

𝑚21

𝜏 , (14.19)

where dots represent derivatives with respect to the time. Using these relations for 𝑣 and 𝜎, weobtain the Lagrangian for the remaining scalar mode (the helicity-0 mode) 𝜏 [437, 200],

ℒ𝜏 =𝑀2

Pl

4

([(4

𝑚24

− 4

𝑚21

)∇𝜏 − 3𝜏

]𝜏 − 2

𝑚22 −𝑚2

3

𝑚44

(∇𝜏)2 (14.20)

+

(4𝑚2

2

𝑚24

− 1

)𝜏∇𝜏 − 3𝑚2

2𝜏2

). (14.21)

In terms of power counting this means that the Lagrangian includes terms of the form 𝑀2Pl𝑚

2∇𝜑𝜑arising from the term going as (𝑚−2

4 −𝑚−21 )∇𝜏𝜏 (where 𝜑 designates the helicity-0 mode which

includes a combination of 𝜎 and 𝑣). Such terms are not present in the Lorentz-invariant Fierz–Paulicase and its non-linear ghost-free extension since 𝑚4 = 𝑚1 in that case, and they play a crucialrole in this Lorentz violating setting.



156 Claudia de Rham

Indeed, in the small mass limit these terms 𝑀2Pl𝑚

2∇𝜑𝜑 dominate over the ones that go as

𝑀2Pl𝑚

4𝜑𝜑 (i.e., the ones present in the Lorentz invariant case). This means that in the small mass

limit, the correct canonical normalization of the helicity-0 mode 𝜑 is not of the form 𝜑 = 𝜑/𝑀Pl𝑚2

but rather 𝜑 = 𝜑/𝑀Pl𝑚√∇, which is crucial in determining the strong coupling scale and the

absence of vDVZ discontinuity:

∙ The new canonical normalization implies a much larger strong coupling scale that goes asΛ2 = (𝑀Pl𝑚)1/2 rather than Λ3 = (𝑀Pl𝑚

2)1/3 as is the case in DGP and ghost-free massivegravity.

∙ Furthermore, in the massless limit the coupling of the helicity-0 mode to the tensor vanishesfasters than some of the Lorentz-violating kinetic interactions in (14.20) (which is scales as

𝑚ℎ𝜕2𝜑). This means that one can take the massless limit 𝑚 → 0 in such a way that thecoupling to the helicity-2 mode disappears and so does the coupling of the helicity-0 mode tomatter (since this coupling arises after de-mixing of the helicity-0 and -2 modes). This impliesthe absence of vDVZ discontinuity in this Lorentz-violating theory despite the presence offive degrees of freedom.

The absence of vDVZ discontinuity and the larger strong coupling scale Λ2 makes this theorymore tractable at small mass scales. We emphasize, however, that the absence of vDVZ discontinu-ity does prevent some sort of Vainshtein mechanism to still come into play since the theory is stillstrongly coupled at the scale Λ2 ≪𝑀Pl. This is similar to what happens for the Lorentz-invariantghost-free theory of massive gravity on AdS (see Section 8.3.6 and [154]). Interestingly, however,the same redressing of the strong coupling scale as in DGP or ghost-free massive gravity was ex-plored in [109] where it was shown that in the vicinity of a localized mass, the strong coupling scalegets redressed in such a way that the weak field approximation remains valid till the Schwarzschildradius of the mass, i.e., exactly as in GR.

In these theories, bounds on the graviton comes from the exponential decay in the Yukawapotential which switches gravity off at the graviton’s Compton wavelength, so the Compton wave-length ought to be larger than the largest gravitational bound states which are of about 5 Mpc,

putting a bound on the graviton mass of𝑚 . 10−30 eV in which case Λ2 ∼(10−4mm

)−1[108, 257].

14.3.4 Cosmology of general massive gravity

The cosmology of general massive gravity was recently studied in [109] and we summarize theirresults in what follows.

In Section 12, we showed how the Bianchi identity in ghost-free massive gravity prevents theexistence of spatially flat FLRW solutions. The situation is similar in general Lorentz violatingtheories of massive gravity unless the function 𝒞 in (14.11) is chosen so as to satisfy the followingrelation when the shift and 𝑛𝑖 vanish [109]

𝐻

(𝒞′ − 1

2𝒞)

= 0 . (14.22)

Choosing a function 𝒞 which satisfies the appropriate condition to allow for FLRW solutions, theFriedmann equation then depends entirely on the function 𝑈(��) also defined in (14.11). In thiscase the graviton potential (14.10) acts as an effective ‘dark fluid’ with respective energy densityand pressure dictated by the function 𝑈 [109]

𝜌eff =𝑚2

4𝑈(��) 𝑝eff =

𝑚2

4(2𝑈 ′(��)− 𝑈(��)) , (14.23)

leading to an effective phantom-like behavior when 2𝑈 ′/𝑈 < 0.



Massive Gravity 157

This solution is stable and healthy as long as the second derivative of 𝒞 satisfies some conditionswhich can easily be accommodated for appropriate functions 𝒞 and 𝑈 .

Expanding 𝑈 in terms of the scale factor for late time 𝑈 =∑𝑛≥0 ��𝑛(𝑎− 1)𝑛 one can use CMB

and BAO data from [2] to put constraints on the first terms of that series [109]

��1

��0= 0.12± 2.1 and

��2

��0< 2± 3 at 95% C.L. (14.24)

Focusing instead on early time cosmology BBN data can similarly be used to constrains the function𝑈 , see [109] for more details.

15 Non-local massive gravity

The ghost-free theory of massive gravity proposed in Part II as well as the Lorentz-violatingtheories of Section 14 require an auxiliary metric. New massive gravity, on the other hand, canbe formulated in a way that requires no mention of an auxiliary metric. Note however that all ofthese theories do break one copy of diffeomorphism invariance, and this occurs in bi-gravity as welland in the zwei dreibein extension of new massive gravity.

One of the motivations of non-local theories of massive gravity is to formulate the theorywithout any reference metric.34 This is the main idea behind the non-local theory of massivegravity introduced in [328].35

Starting with the linearized equation about flat space-time of the Fierz–Pauli theory

𝛿𝐺𝜇𝜈 −1

2𝑚2 (ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈) = 8𝜋𝐺𝑇𝜇𝜈 , (15.1)

where 𝛿𝐺𝜇𝜈 = ℰ𝛼𝛽𝜇𝜈 ℎ𝛼𝛽 is the linearized Einstein tensor, this modified Einstein equation can be‘covariantized’ so as to be valid about for any background metric. The linearized Einstein tensor𝛿𝐺𝜇𝜈 gets immediately covariantized to the full Einstein tensor 𝐺𝜇𝜈 . The mass term, on the otherhand, is more subtle and involves non-local operators. Its covariantization can take different forms,and the ones considered in the literature that do not involve a reference metric are

1

2(ℎ𝜇𝜈 − ℎ𝜂𝜇𝜈) −→

{(2−1𝑔 𝐺𝜇𝜈

)𝑇Ref. [328]

38

(𝑔𝜇𝜈2

−1𝑔 𝑅

)𝑇Refs. [395, 229, 228]

, (15.2)

where 2𝑔 is the covariant d’Alembertian 2𝑔 = 𝑔𝜇𝜈∇𝜇∇𝜈 and 2−1𝑔 represents the retarded propaga-

tor. One could also consider a linear combination of both possibilities. Furthermore, any of theseterms could also be implemented by additional terms that vanish on flat space, but one shouldtake great care in ensuring that they do not propagate additional degrees of freedom (and ghosts).

Following [328] we use the notation where 𝑇 designates the transverse part of a tensor. For anytensor 𝑆𝜇𝜈 ,

𝑆𝜇𝜈 = 𝑆𝑇𝜇𝜈 +∇(𝜇𝑆𝜈) , (15.3)

with ∇𝜇𝑆𝑇𝜇𝜈 = 0. In flat space we can infer the relation [328]

𝑆𝑇𝜇𝜈 = 𝑆𝜇𝜈 −2

2𝜕(𝜇𝜕

𝛼𝑆𝜈)𝛼 +1

22𝜕𝜇𝜕𝜈𝜕

𝛼𝜕𝛽𝑆𝛼𝛽 . (15.4)

34 Notice that even if massive gravity is formulated without the need of a reference metric, this does not changethe fact that one copy of diffeomorphism invariance in broken leading to additional degrees of freedom as is the casein new massive gravity.

35 See also [335, 195, 49, 50, 51] for other ghost-free non-local modifications of gravity, but where the graviton ismassless.



158 Claudia de Rham

The theory propagates what looks like a ghost-like instability irrespectively of the exact formu-lation chosen in (15.2). However, it was recently argued that the would-be ghost is not a radiativedegree of freedom and therefore does not lead to any vacuum decay. It remains an open questionof whether the would be ghost can be avoided in the full nonlinear theory.

The cosmology of this model was studied in [395, 228]. The new contribution (15.2) in theEinstein equation can play the role of dark energy. Taking the second formulation of (15.2) andsetting the graviton mass to 𝑚 ≃ 0.67𝐻0, where 𝐻0 is the Hubble parameter today, reproduces theobserved amount of dark energy. The mass term acts as a dark fluid with effective time-dependentequation of state 𝜔eff(𝑎) ≃ −1.04−0.02(1−𝑎), where 𝑎 is the scale factor, and is thus phantom-like.

Since this theory is formulated at the level of the equations of motion and not at the level ofthe action and since it includes non-local operators it ought to be thought as an effective classicaltheory. These equations of motion should not be used to get some insight on the quantum natureof the theory nor on its quantum stability. New physics would kick in when quantum correctionsought to be taken into account. It remains an open question at the moment of how to embednonlocal massive gravity into a consistent quantum effective field theory.

Notice, however, that an action principle was proposed in Ref. [402], (focusing on four dimen-sions),

𝑆 =

∫d4𝑥√−𝑔[𝑀2

Pl

2𝑅+ ��+

𝑀2Pl

2𝑀2𝑅𝜇𝜈ℎ

(− 2

𝑀2

)𝐺𝜇𝜈

], (15.5)

where the function ℎ is defined as

ℎ(𝑧) =2+𝑚2

2

1

𝑧|𝑝𝑠(𝑧)|𝑒

12Γ(0,𝑝

2𝑠(𝑧))+

12𝛾𝐸 , (15.6)

where 𝛾𝐸 = 0.577216 is the Euler’s constant, Γ(𝑏, 𝑧) =∫∞𝑧𝑡𝑏−1𝑒−𝑡 is the incomplete gamma

function, 𝑠 is a integer 𝑠 > 3 and 𝑝𝑠(𝑧) is a real polynomial of rank 𝑠. Upon deriving the equationsof motion we recover the non-local massive gravity Einstein equation presented above [402],

𝐺𝜇𝜈 +𝑚2

2𝐺𝜇𝜈 =

𝑀2Pl

2𝑇𝜇𝜈 , (15.7)

up to order 𝑅2 corrections. We point out however that in this action derivation principle theoperator 2−1 likely correspond to a symmetrized Green’s function, while in (15.2) causality requires2−1 to represent the retarded one.

We stress, however, that this theory should be considered as a classical theory uniquely andnot be quantized. It is an interesting question of whether or not the ghost reappears when consid-ering quantum fluctuations like the ones that seed any cosmological perturbations. We emphasizefor instance that when dealing with any cosmological perturbations, these perturbations have aquantum origin and it is important to rely of a theory that can be quantized to describe them.



Massive Gravity 159

16 Outlook

The past decade has witnessed a revival of interest in massive gravity as a potential alternativeto GR. The original theoretical obstacles that came in the way of deriving a consistent theory ofmassive gravity have now been overcome, but with them comes a new set of challenges that will bedecisive in establishing the viability of such theories. The presence of a low strong coupling scale onwhich the Vainshtein mechanism relies has opened the door to a new way of thinking about thesetypes of effective field theories. At the moment, it is yet unclear whether these types of theoriescould lead to an alternative to UV completion. The superluminalities that also arise in manycases with the Vainshtein mechanism should also be understood in more depth. At the moment,its real implications are not well understood and no case of true acausality has been shown to bepresent within the regime of validity of the theory. Finally, the difficulty in finding fully-fledgedcosmological and black-hole solutions in many of these theories (both in ghost-free massive gravityand bi-gravity, and in other extensions or related models such as cascading gravity) makes their fullphenomenology still evasive. Nevertheless, the well understood decoupling limits of these modelscan be used to say a great deal about phenomenology without going into the complications of thefull theories. These represent many open questions in massive gravity, which reflect the fact thatthe field is yet extremely young and many developments are still in progress.

Acknowledgments

CdR wishes to thank Denis Comelli, Matteo Fasiello, Gregory Gabadadze, Daniel Grumiller, KurtHinterbichler, Andrew Matas, Shinji Mukohyama, Nicholas Ondo, Luigi Pilo and especially AndrewTolley for useful discussions. CdR is supported by Department of Energy grant DE-SC0009946.



160 Claudia de Rham

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